Oo Haskell

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Haskell’s overlooked object system — 10 September 2005 — Oleg Kiselyov Fleet Numerical Meteorology and Oceanography Center, Monterey, CA

Ralf L¨ammel Microsoft Corp., Redmond, WA

Abstract Haskell provides type-class-bounded and parametric polymorphism as opposed to subtype polymorphism of object-oriented languages such as Java and OCaml. It is a contentious question whether Haskell 98 without extensions, or with common extensions, or with new extensions can fully support conventional object-oriented programming with encapsulation, mutable state, inheritance, overriding, statically checked implicit and explicit subtyping, and so on. In a first phase, we demonstrate how far we can get with object-oriented functional programming, if we restrict ourselves to plain Haskell 98. In the second and major phase, we systematically substantiate that Haskell 98, with some common extensions, supports all the conventional OO features plus more advanced ones, including first-class lexically scoped classes, implicitly polymorphic classes, flexible multiple inheritance, safe downcasts and safe co-variant arguments. Haskell indeed can support width and depth, structural and nominal subtyping. We address the particular challenge to preserve Haskell’s type inference even for objects and object-operating functions. Advanced type inference is a strength of Haskell that is worth preserving. Many of the features we get “for free”: the type system of Haskell turns out to be a great help and a guide rather than a hindrance. The OO features are introduced in Haskell as the OOHaskell library, non-trivially based on the HList library of extensible polymorphic records with first-class labels and subtyping. The library sample code, which is patterned after the examples found in OO textbooks and programming language tutorials, including the OCaml object tutorial, demonstrates that OO code translates into OOHaskell in an intuition-preserving way: essentially expression-by-expression, without requiring global transformations. OOHaskell lends itself as a sandbox for typed OO language design. Keywords: Object-oriented functional programming, Object type inference, Typed objectoriented language design, Heterogeneous collections, ML-ART, Mutable objects, TypeClass-based programming, Haskell, Haskell 98, Structural subtyping, Duck typing, Nominal subtyping, Width subtyping, Deep subtyping, Co-variance

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O. Kiselyov and R. L¨ ammel Contents

1 Introduction 2 The folklore ‘shapes’ example 2.1 C++ reference encoding 2.2 OOHaskell encoding 2.3 Discussion of the example 2.3.1 Classes vs. interfaces 2.3.2 Extensibility and encapsulation 2.3.3 Subtyping technicalities 3 Alternative Haskell encodings 3.1 Map subtype hierarchy to an algebraic datatype 3.2 Map object data to tail-polymorphic record types 3.3 Functional objects, again with tail polymorphism 3.4 Mutable objects, again with tail polymorphism 3.5 Subtypes as composed record types with overloading 3.6 Variation — existential quantification 3.7 Variation — heterogeneous collections 4 Type-agnostic OOHaskell idioms 4.1 Objects as records 4.1.1 First-class labels 4.1.2 Mutable variables 4.1.3 HList records 4.1.4 OO test cases 4.2 Object generators 4.2.1 Constructor arguments 4.2.2 Construction-time computations 4.2.3 Implicitly polymorphic classes 4.2.4 Nested object generators 4.2.5 Open recursion 4.2.6 Instantiation checking 4.3 Reuse techniques 4.3.1 Single inheritance with extension 4.3.2 Class-polymorphic functionality 4.3.3 Single inheritance with override 4.3.4 Orphan methods 4.3.5 Flexible reuse schemes 4.4 Safe value recursion 5 Type-perceptive OOHaskell idioms 5.1 Semi-implicit upcasting 5.2 Narrow to a fixed type 5.3 Self-returning methods 5.4 Casts based on dynamics 5.5 Casts based on unions 5.6 Explicit type constraints 5.7 Nominal subtyping 5.8 Iso-recursive types 5.9 Width and depth subtyping 5.10 Co-variant method arguments 5.11 Anti-patterns for subtyping 6 Discussion 6.1 Usability issues 6.1.1 Usability of inferred types 6.1.2 Usability of type errors 6.1.3 Efficiency of object encoding 6.2 Related work 6.2.1 The ML-ART object encoding 6.2.2 Haskell language extensions 6.2.3 Object encodings for Haskell 6.3 More future work 7 Concluding remarks References

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1 Introduction The topic of object-oriented programming in the functional language Haskell is raised time and again on programming language mailing lists, on programming tutorial websites, and in verbal communication at programming language conferences with remarkable intensity. Dedicated OO Haskell language extensions have been proposed; specific OO idioms have been encoded in Haskell (Hughes & Sparud, 1995; Gaster & Jones, 1996; Finne et al., 1999; Shields & Peyton Jones, 2001; Nordlander, 2002; Bayley, 2005). The interest in this topic is not at all restricted to Haskell researchers and practitioners since there is a fundamental and unsettled question — a question that is addressed in the present paper:1 What is the relation between type-class-bounded and subtype polymorphism? In this research context, we specifically (and emphatically) restrict ourselves to the existing Haskell language (Haskell 98 and common extensions where necessary), i.e., no new Haskell extensions are to be proposed. As we will substantiate, this restriction is adequate, as it allows us to deliver a meaningful and momentous answer to the aforementioned question. At a more detailed level, we offer the following motivation for research on OO programming in Haskell: • In an intellectual sense, one may wonder whether Haskell’s advanced type system is expressive enough to model object types, inheritance, subtyping, virtual methods, etc. No general, conclusive result has been available so far. • In a practical sense, one may wonder whether we can faithfully transport imperative OO designs from, say, C#, C++, Eiffel, Java, VB to Haskell — without totally rewriting the design and without foreign-language interfacing. • From a language design perspective, Haskell has a strong record in prototyping semantics and encoding abstraction mechanisms, but one may wonder whether Haskell can perhaps even serve as a sandbox for design of typed object-oriented languages so that one can play with new ideas without the immediate need to write or modify a compiler. • In an educational sense, one may wonder whether more or less advanced functional and object-oriented programmers can improve their understanding of Haskell’s type system and OO concepts by looking into the pros and cons of different OO encoding options in Haskell. This paper delivers substantiated, positive answers to these questions. We describe OOHaskell — a Haskell-based library for (as of today: imperative) OO 1

On a more anecdotal account, we have collected informative pointers to mailing list discussions, which document the unsettled understanding of OO programming in Haskell and the relation between OO classes and Haskell’s type classes: http://www.cs.mu.oz.au/research/mercury/ mailing-lists/mercury-users/mercury-users.0105/0051.html, http://www.talkaboutprogramming. com/group/comp.lang.functional/messages/47728.html, http://www.haskell.org/pipermail/haskell/ 2003-December/013238.html, http://www.haskell.org/pipermail/haskell-cafe/2004-June/006207.html, http://www.haskell.org//pipermail/haskell/2004-June/014164.html

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programming in Haskell. OOHaskell delivers Haskell’s “overlooked” object system. The key to this result is a good deal of exploitation of Haskell’s advanced type system combined with a careful identification of a suitable object encoding. We instantiate and enhance existing encoding techniques (such as (Pierce & Turner, 1994; R´emy, 1994; Abadi & Cardelli, 1996)) aiming at a practical object system that blends well with the host language — Haskell. We take advantage of our previous work on heterogeneous collections (Kiselyov et al., 2004) (the HList library). More generally, we put type-class-based or type-level programming to work (Hallgren, 2001; McBride, 2002; Neubauer et al., 2002; Neubauer et al., 2001). The simplified story is the following: - Classes are represented as functions that are in fact object generators. - State is maintained through mutable variables allocated by object generators. - Objects are represented as records of closures with a component for each method. - Methods are monadic functions that can access state and self. - We use HList’s record calculus (extensible records, up-casts, etc.). - We use type-class-based functionality to program the object typing rules. To deliver a faithful, convenient and comprehensive object system, several techniques had to be discovered and combined. Proper effort was needed to preserve Haskell’s type inference for OO programming idioms (as opposed to explicit type declarations or type constraints for classes, methods, and up-casts). The obtained result, OOHaskell, delivers an amount of polymorphism and type inference that is unprecedented. Proper effort was also needed in order to deploy value recursion for closing object generators. Achieving safety of this approach was a known challenge (R´emy, 1994). In order to fully appreciate the object system of OOHaskell, we also review less sophisticated, less favourable encoding alternatives. Not only OOHaskell provides the conventional OO idioms; we have also language-engineered several features that are either bleeding-edge or unattainable in mainstream OO languages: for example, first-class classes and class closures; statically type-checked collection classes with bounded polymorphism of implicit collection arguments; multiple inheritance with user-controlled sharing; safe co-variant argument subtyping. It is remarkable that these and more familiar object-oriented features are not introduced by fiat — we get them for free. For example, the type of a collection with bounded polymorphism of elements is inferred automatically by the compiler. Also, abstract classes are uninstantiatable not because we say so but because the program will not typecheck otherwise. Co- and contra-variant subtyping rules and the safety conditions for the co-variant method argument types are checked automatically without any programming on our part. These facts suggest that (OO)Haskell lends itself as prime environment for typed object-oriented language design.

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Road-map of this paper • Sec. 2: We encode a tutorial OO example both in C++ and OOHaskell. • Sec. 3: We review alternative object encodings in Haskell 98 and beyond. • Sec. 4 and Sec. 5: We describe all OOHaskell idioms. The first part focuses on idioms where subtyping and object types do not surface the OO program code. The second part covers all technical details of subtyping including casts and variance properties. • Sec. 6: We discuss usability issues, related work and future work. • Sec. 7: We conclude the paper. The main sections, Sec. 4 and Sec. 5, are written in tutorial style, as to ease digestion of all techniques, as to encourage OO programming and OO language design experiments. There is an extended source distribution available.2 2 The folklore ‘shapes’ example One of the main goals of this paper is to be able to represent the conventional OO code, in as straightforward way as possible. The implementation of our system may be not for the feeble at heart — however, the user of the system must be able to write conventional OO code without understanding the complexity of the implementation. Throughout the paper, we illustrate OOHaskell with a series of practical examples as they are commonly found in OO textbooks and programming language tutorials. In this section, we begin with the so-called ‘shapes’ example. We face a type for ‘shapes’ and two subtypes for ‘rectangles’ and ‘circles’; see Fig. 1. Shapes maintain coordinates as state. Shapes can be moved around and drawn. The exercise shall be to place objects of different kinds of shapes in a collection and to iterate over them as to draw the shapes. It turns out that this example is a crisp OO benchmark.3 2.1 C++ reference encoding The type of shapes can be defined as a C++ class as follows: class Shape { public: // Constructor method Shape(int newx, int newy) { x = newx; y = newy; } 2

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The source code can be downloaded at http://www.cwi.nl/~ralf/OOHaskell/, and it is subject to a very liberal license (MIT/X11 style). As of writing, the actual code commits to a few specific extensions of the GHC implementation of Haskell — for reasons of convenience. In principle, Haskell 98 + multi-parameter classes with functional dependencies is sufficient. The ‘shapes’ problem has been designed by Jim Weirich and deeply explored by him and Chris Rathman. See the multi-lingual collection ‘OO Example Code’ by Jim Weirich at http:// onestepback.org/articles/poly/; see also an even heavier collection ‘OO Shape Examples’ by Chris Rathman at http://www.angelfire.com/tx4/cus/shapes/.

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Shape

x,y: Int + moveTo + rMoveTo + draw

Rectangle

Circle

width,height: Int

radius: Int

+ draw

+ draw

Fig. 1. Shapes with state and a subtype-specific draw method

// Accessors int getX() { return int getY() { return void setX(int newx) void setY(int newy)

x; } y; } { x = newx; } { y = newy; }

// Move shape position void moveTo(int newx, int newy) { x = newx; y = newy; } // Move shape relatively void rMoveTo(int deltax, int deltay) { moveTo(getX() + deltax, getY() + deltay); } // An abstract draw method virtual void draw() = 0; // Private data private: int x; int y; }

The x, y coordinates are private, but they can be accessed through getters and setters. The methods for accessing and moving shapes are inherited by the subclasses of Shape. The draw method is virtual and even abstract; hence concrete subclasses must implement draw. The subclass Rectangle is derived as follows:

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class Rectangle: public Shape { public: // Constructor method Rectangle(int newx, int newy, int newwidth, int newheight) : Shape(newx, newy) { width = newwidth; height = newheight; } // Accessors int getWidth() { return width; } int getHeight() { return height; } void setWidth(int newwidth) { width = newwidth; } void setHeight(int newheight) { height = newheight; } // Implementation of the abstract draw method void draw() { cout << "Drawing a Rectangle at:(" << getX() << "," << getY() << "), width " << getWidth() << ", height " << getHeight() << endl; } // Additional private data private: int width; int height; };

For brevity, we elide the similar derivation of the subclass Circle: class Circle : public Shape { Circle(int newx, int newy, int newradius) : Shape(newx, newy) { ... } ... }

The following code block constructs different shape objects and invokes their methods. More precisely, we place two shapes of different kinds in an array, scribble, and then loop over it to draw and move the shape objects: Shape *scribble[2]; scribble[0] = new Rectangle(10, 20, 5, 6); scribble[1] = new Circle(15, 25, 8); for (int i = 0; i < 2; i++) { scribble[i]->draw(); scribble[i]->rMoveTo(100, 100); scribble[i]->draw(); }

The loop over scribble exercises subtyping polymorphism: the actually executed implementation of the draw method differs per element in the array. The program run produces the following output — due to the logging-like implementations of the draw method:

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Drawing Drawing Drawing Drawing

a a a a

Rectangle at:(10,20), width 5, height 6 Rectangle at:(110,120), width 5, height 6 Circle at:(15,25), radius 8 Circle at:(115,125), radius 8

2.2 OOHaskell encoding We now show an OOHaskell encoding, which happens to pleasantly mimic the C++ encoding, while any remaining deviations are appreciated. Most notably, we are going to leverage type inference: we will not define any type. The code shall be fully statically typed nevertheless. Here is the OOHaskell rendering of the shape class: -- Object generator for shapes shape newx newy self = do -- Create references for private state x <- newIORef newx y <- newIORef newy -- Return object as record of methods returnIO $ getX .=. readIORef x .*. getY .=. readIORef y .*. setX .=. writeIORef x .*. setY .=. writeIORef y .*. moveTo .=. (\newx newy -> do (self # setX) newx (self # setY) newy ) .*. rMoveTo .=. (\deltax deltay -> do x <- self # getX y <- self # getY (self # moveTo) (x + deltax) (y + deltay) ) .*. emptyRecord

Classes become functions that take constructor arguments plus a self reference and that return a computation whose result is the new object — a record of methods including getters and setters. We can invoke methods of the same object through self; cf. the method invocation self # getX and others. (The infix operator # denotes method invocation.) Our objects are mutable, implemented via IORef. (STRef also suffices.) Since most OO systems in practical use have mutable state, OOHaskell does not (yet) offer functional objects, which are known to be challenging on their own. We defer functional objects to future work. We use the extensible records of the HList library (Kiselyov et al., 2004), hence: • • • •

emptyRecord denotes what the name promises, (.*.) stands for (right-associative) record extension, (.=.) is record-component construction: label .=. value, Labels are defined according to a trivial scheme, to be explained later.

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The abstract draw method is not mentioned in the OOHaskell code because it is not used in any other method, neither did we dare declaring its type. As a side effect, the object generator shape is instantiatable whereas the explicit declaration of the abstract draw method made the C++ class Shape uninstantiatable. We will later show how to add similar declarations in OOHaskell. We continue with the OOHaskell code for the shapes example. -- Object generator for rectangles rectangle newx newy width height self = do -- Invoke object generator of superclass super <- shape newx newy self -- Create references for extended state w <- newIORef width h <- newIORef height -- Return object as returnIO $ getWidth .=. .*. getHeight .=. .*. setWidth .=. .*. setHeight .=.

record of methods readIORef readIORef (\neww -> (\newh ->

w h writeIORef w neww) writeIORef h newh)

.*. draw .=. do -- Implementation of the abstract draw method putStr "Drawing a Rectangle at:(" << self # getX << ls "," << self # getY << ls "), width " << self # getWidth << ls ", height " << self # getHeight << ls "\n" -- Rectangle records start from shape records .*. super

This snippet illustrates the essence of inheritance in OOHaskell. Object generation for the supertype is made part of the monadic sequence that defines object generation for the subtype; self is passed from the subtype to the supertype. Subtype records are derived from supertype records through record extension (or potentially also through record updates when overrides are to be modelled). As in the C++ case, we elide the derivation of the object generators for circles: circle newx newy newradius self = do super <- shape newx newy self ... returnIO ... .*. super

Ultimately, here is the OOHaskell rendering of the ‘scribble loop’: -- Object construction and invocation as a monadic sequence myOOP = do -- Construct objects s1 <- mfix (rectangle (10::Int) (20::Int) 5 6) s2 <- mfix (circle (15::Int) 25 8)

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O. Kiselyov and R. L¨ ammel -- Create a homogeneous list of different shapes let scribble = consLub s1 (consLub s2 nilLub) -- Loop over list with normal monadic map mapM_ (\shape -> do shape # draw (shape # rMoveTo) 100 100 shape # draw) scribble

The use of mfix (an analogue of the new in C++) reflects that object generators take ‘self’ and construct (part of) it. Open recursion enables inheritance. The let scribble . . . binding is noteworthy. We cannot directly place rectangles and circles in a normal Haskell list — the following cannot possibly type check: let scribble = [s1,s2] -- s1 and s2 are of different types!

We have to homogenise the types of s1 and s2 when forming a Haskell list. To this end, we use special list constructors nilLub and consLub as opposed to the normal list constructors [] and (:). These new constructors coerce the list elements to the least-upper bound type of all the element types. Incidentally, if the ‘intersection’ of the types of the objects s1 and s2 does not include the methods that are invoked later (i.e., draw and rMoveTo), we get a static type error which literally says so. As a result, the original for-loop can be carried out in the native Haskell way: a normal (monadic) list map over a normal Haskell list of shapes. Hence, we have exercised a faithful model of subtype polymorphism, which also allows for (almost) implicit subtyping. OOHaskell provides several subtyping models, as we will study later. 2.3 Discussion of the example 2.3.1 Classes vs. interfaces The C++ code should not be misunderstood to suggest that class inheritance is the only OO design option for the shapes hierarchy. In a Java-like language, one may want to model Shape as an interface, say, IShape, with Rectangle and Circle as classes implementing this interface. This design would not allow us to reuse the implementations of the accessors and the move methods. So one may want to combine interface polymorphism and class inheritance. That is, the classes Rectangle and Circle will be rooted by an additional implementation class for shapes, say Shape, which hosts implementations shared among different shape classes — incidentally a part of the IShape interface. The remainder of the IShape interface, namely the draw method in our example, would be implemented in Rectangle and Circle. More generally, OO designs that employ interface polymorphism alone are rare, so we need to provide encodings for both OO interface polymorphism and OO class inheritance in OOHaskell. One may say that the former mechanism is essentially covered by Haskell’s type classes (modulo the fact that we would still need an object encoding). The latter mechanism is specifically covered by original HList and OOHaskell contributions: structural subtyping polymorphism for object types, based on polymorphic extensible records and programmable subtyping constraints. (Sec. 5.7 discusses nominal object types in OOHaskell)˙

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2.3.2 Extensibility and encapsulation Both the C++ encoding and the OOHaskell encoding of the shapes example are faithful to the encapsulation premise as well as the extensibility premise of the OO paradigm. An object encapsulates both data (‘state’) and methods (‘behaviour’). One may add new kinds of shapes without rewriting (or, perhaps, even re-compiling) existing code. Both premises are the subject of an unsettled debate in the programming language community, especially with regards to functional programming. The basic OO paradigm has been criticised (Zenger & Odersky, 2004) for its over-emphasis of extensibility in the subtyping dimension and for its neglect of other dimensions such as the addition of new functions into a pre-existing subtyping hierarchy. While we agree with this overall criticism, we avoid the debate in this paper. We simply want OOHaskell to provide an object encoding that is compatible with the established OO paradigm. (Incidentally, some of the non-encapsulation-based encodings in Sec. 3 show that Haskell supports extensibility in both the data and the functionality dimension.) 2.3.3 Subtyping technicalities The “scribble loop” is by no means a contrived scenario. It is a faithful instance of the ubiquitous composite design pattern (Gamma et al., 1994). In terms of expressiveness and typing challenges, this sort of loop over an array of shapes of different kinds forces us to explore the tension between implicit and explicit subtyping. As we will discuss, it is relatively straightforward to use type-class-bounded polymorphism to represent subtype constraints. It is however less straightforward to accumulate entities of different subtypes in the same collection. With explicit subtyping (e.g., by wrapping in a properly constrained existential envelope) the burden would be on the side of the programmer. A key challenge for OOHaskell was to make subtyping (almost) implicit in all the cases, where a OO programmer would expect it. This is a particular area in which OOHaskell goes beyond OCaml (Leroy et al. , 2004)— the de-facto leading strongly typed functional object-oriented language. OOHaskell provides a range of subtyping notions, including one that even allows for safe downcasts for object types. This is again something that has not been achieved in OCaml to date. 3 Alternative Haskell encodings OOHaskell goes particularly far in providing an object system, when compared to conservative Haskell programming knowledge. To this end, we put type-classbased or type-level programming to work. In this section, we will review more conservative object encodings with their characteristics and limitations. All of them require boilerplate code from the programmer. Some of the ‘conservative’ encodings to come are nevertheless involved and enlightening. In fact, the full spectrum of encodings has not been documented before — certainly not in a Haskell context. So we reckon that their detailed analysis

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makes a useful contribution. Furthermore, several of the discussed techniques are actually used in OOHaskell, where some of them are simply generalised through the advanced use of Haskell’s type classes. Hence, the present section is an incremental preparation for the main sections Sec. 4 and Sec. 5. For most of this section, we limit ourselves to Haskell 98. (By contrast, OOHaskell requires several common Haskell 98 extensions.) Towards the end of the section, we will investigate the value of dismissing this restriction. 3.1 Map subtype hierarchy to an algebraic datatype We begin with a trivial and concise encoding. Its distinguishing characteristic is extreme simplicity.4 It uses only basic Haskell 98 idioms. The encoding is also seriously limited, lacking extensibility with regard to new forms of shapes (cf. Sec. 2.3.2). We define an algebraic datatype for shapes, where each kind of shape amounts to a constructor declaration. For readability, we use labelled fields instead of unlabelled constructor components. data Shape = Rectangle { , , ,

getX getY getWidth getHeight

:: :: :: ::

Int Int Int Int }

| Circle { getX :: Int , getY :: Int , getRadius :: Int }

Both constructor declarations involve labelled fields for the (x, y) position of a shape. While this reusability dimension is not emphasised at the datatype level, we can easily define reusable setters for the position. (There are some issues regarding type safety, which we will address later.) For instance: setX :: Int -> Shape -> Shape setX i s = s { getX = i }

We can also define setters for Rectangle- and Circle-specific fields. For instance: setWidth :: Int -> Shape -> Shape setWidth i s = s { getWidth = i }

It is also straightforward to define functions for moving around shapes: moveTo :: Int -> Int -> Shape -> Shape moveTo x y = setY y . setX x rMoveTo :: Int rMoveTo deltax where x = getX s + y = getY s + 4

-> Int -> Shape -> Shape deltay s = moveTo x y s deltax deltay

Thanks to Lennart Augustsson for pointing out this line of encoding. Cf. http://www.haskell.org/pipermail/haskell/2005-June/016061.html

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The function for drawing shapes properly discriminates on the kind of shapes. That is, there is one equation per kind of shape. Subtype polymorphism reduces to pattern matching, so to say: draw :: Shape -> IO () draw s@(Rectangle _ _ _ _) = putStrLn ("Drawing a Rectangle at:(" ++ (show (getX s)) ++ "," ++ (show (getY s)) ++ "), width " ++ (show (getWidth s)) ++ ", height " ++ (show (getHeight s))) draw s@(Circle _ _ _) = putStrLn ("Drawing a Circle at:(" ++ (show (getX s)) ++ "," ++ (show (getY s)) ++ "), radius " ++ (show (getRadius s)))

With this encoding, it is trivial to build a collection of shapes of different kinds and to iterate over it such that each shape is drawn and moved (and drawn again): main = do let scribble = [ Rectangle 10 20 5 6 , Circle 15 25 8 ] mapM_ ( \x -> do draw x draw (rMoveTo 100 100 x)) scribble

Assessment of the encoding • The encoding ignores the encapsulation premise of the OO paradigm. • The foremost weakness of the encoding is the lack of extensibility. The addition of a new kind of shape would require re-compilation of all code; it would also require amendments of existing definitions or declarations: the datatype declaration Shape and the function definition draw. • A related weakness is that the overall scheme does not suggest a way of dealing with virtual methods: introduce the type of a method for a base type potentially with an implementation; define or override the method for a subtype. We would need a scheme that offers (explicit and implicit) open recursion for datatypes and functions defined on them. • The setters setX and setY happen to be total because all constructors end up defining labelled fields getX and getY. The type system does not prevent us from forgetting those labels for some constructor. It is relatively easy to

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O. Kiselyov and R. L¨ ammel resolve this issue to the slight disadvantage of conciseness. (For instance, we may avoid labelling entirely, and use pattern matching instead. We may also compose together rectangles and circles from common shape data and deltas.) • The use of a single algebraic datatype Shape implies that Rectangle- and Circle-specific functions cannot be defined as total functions. Such biased functions, e.g., setWidth, are only defined for certain constructors. Once we go beyond the simple-minded encoding model of this section, it will be possible to increase type safety by making type distinctions for different kinds of shapes, but then we will also encounter the challenge of subtype polymorphism.

3.2 Map object data to tail-polymorphic record types There is a folklore technique for encoding extensible records (Burton, 1990) that we can use to model the shapes hierarchy in Haskell 98. Simple type classes let us implement virtual methods. We meet the remaining challenge of placing different shapes into one list by making different subtypes homogeneous through embedding shape subtypes into a union type (Haskell’s Either). We begin with a datatype for extensible shapes; cf. shapeTail: data Shape w = Shape { getX :: Int , getY :: Int , shapeTail :: w }

For convenience, we also provide a constructor for shapes: shape x y w = Shape { getX = x , getY = y , shapeTail = w }

We can define setters and movers once and for all for all possible extensions of Shape by simply leaving the extension type parametric. The actual equations are literally the same as in the previous section; so we only show the (different) parametrically polymorphic types: setX :: Int -> Shape w -> Shape w setY :: Int -> Shape w -> Shape w moveTo :: Int -> Int -> Shape w -> Shape w rMoveTo :: Int -> Int -> Shape w -> Shape w

The presence of the type variable w expresses that the earlier definitions on Shape . . . can clearly be instantiated to all subtypes of Shape. The draw function must be placed in a dedicated type class, Draw, because we anticipate the need to provide type-specific implementations of draw. (One may compare this style with C++ where one explicitly declares a method to be (pure) virtual.) class Draw w where draw :: Shape w -> IO ()

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Shape extensions for rectangles and circles are built according to a common scheme. We only show the details for rectangles. We begin with the definition of the “data delta” contributed by rectangles; each such delta is again polymorphic in its tail. data RectangleDelta w = RectangleDelta { getWidth :: Int , getHeight :: Int , rectangleTail :: w }

We define the type of rectangles as an instance of Shape: type Rectangle w = Shape (RectangleDelta w)

For convenience, we provide a constructor for rectangles. Here we fix the tail of the rectangle delta to (). (We could still further instantiate Rectangle and define new constructors later, if necessary.) rectangle x y w h = shape x y $ RectangleDelta { getWidth = w , getHeight = h , rectangleTail = () }

The definition of rectangle-specific setters involves nested record manipulation: setHeight :: Int -> Rectangle w -> Rectangle w setHeight i s = s { shapeTail = (shapeTail s) { getHeight = i } } setWidth :: Int -> Rectangle w -> Rectangle w setWidth i s = s { shapeTail = (shapeTail s) { getWidth = i } }

The rectangle-specific draw function is defined through a Draw instance: instance Draw (RectangleDelta w) where draw s = putStrLn ("Drawing a Rectangle at:(" ++ (show (getX s)) ++ "," ++ (show (getY s)) ++ "), width " ++ (show (getWidth (shapeTail s))) ++ ", height " ++ (show (getHeight (shapeTail s))))

The difficult part is the ‘scribble loop’. We cannot easily form a collection of shapes of different kinds. For instance, the following attempt will not type-check: -- Wrong! There is no homogeneous element type. let scribble = [ rectangle 10 20 5 6 , circle 15 25 8 ]

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There is a relatively simple technique to make rectangles and circles homogeneous within the scope of the scribble list and its clients. We have to establish a union type for the different kinds of shapes.5 Using an appropriate helper, tagShape, for embedding shapes into a union type (Haskell’s Either), we may construct a homogeneous collection as follows: let scribble = [ tagShape Left (rectangle 10 20 5 6) , tagShape Right (circle 15 25 8) ]

The boilerplate operation for embedding is trivially defined as follows. tagShape :: (w -> w’) -> Shape w -> Shape w’ tagShape f s = s { shapeTail = f (shapeTail s) }

Embedding (or tagging) clearly does not disturb the reusable definitions of functions on Shape w. However, the loop over scribble refers to the draw operation, which is defined for RectangleDelta and CircleDelta, but not for the union over these two types. We have to provide a trivial boilerplate for generalising draw: instance (Draw a, Draw b) => Draw (Either a b) where draw = eitherShape draw draw

(This instance actually suffices for arbitrarily nested unions of Shape subtypes.) Here, eitherShape is variation on the normal fold operation for unions, i.e., either. We discriminate on the Left vs. Right cases for the tail of a shape datum. This boilerplate operation is independent of draw, but specific to Shape. eitherShape :: (Shape w -> t) -> (Shape w’ -> t) -> Shape (Either w w’) -> t eitherShape f g s = case shapeTail s of (Left s’) -> f (s { shapeTail = s’ }) (Right s’) -> g (s { shapeTail = s’ })

The Draw instance for Either makes it clear that we use the union type as an intersection type. We may only invoke a method on the union only if we may invoke the method on either branch of the union. The instance constraints make that fact obvious. Assessment of the encoding • Again, the encoding ignores the encapsulation premise of the OO paradigm: methods are not encapsulated along with the data. • The encoding does not have the basic extensibility problem of the previous section. We can introduce new kinds of shapes without rewriting and recompiling type-specific code.

5

Haskell 98 supports unions in the prelude: with the type name Either, and the two constructors Left and Right for the branches of the union.

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• Some patterns of subtype-polymorphic code may require revision, though. For instance all program points that insert into a subtype-polymorphic collection or that downcast must agree on the formation of the union type over specific subtypes. If a new subtype must be covered, then the scattered applications of embedding operations must be revised. • We fail to put Haskell’s type inference to work as far as object types are concerned. We end up defining explicit datatypes for all encoded classes. This is acceptable from a mainstream OO point of view since nominal types (i.e., explicit types) dominate the OO paradigm. However, in Haskell, we would like to do better by allowing for inference of structural class and interface types. All subsequent encodings of this section will share this problem. (By contrast, OOHaskell provides full structural type inference.) • It is annoying enough that the formation of a subtype-polymorphic collection requires explicit tagging of all elements; cf. Left and Right. What is worse, the tagging is done on the delta position of Shape. This makes the scheme noncompositional: Each new base class requires its own functions like tagShape and eitherShape. • The encoding of final and virtual methods differs essentially. The former are encoded as parametric polymorphic functions parameterised in the extension type. Virtual methods are encoded as type-class-bounded polymorphic functions overloaded in the extension type. Changing a final method into virtual or vice versa triggers code rewriting. This may be overcome by making all methods virtual (and using default type class methods to reuse implementations). However, this bias will increase the amount of boilerplate code such as the instances for Either. • The subtyping hierarchy leaks into the encoding of subtype-specific accessors; cf. setWidth. The derivation chain from a base type shows up as nesting depth in the record access pattern. One may factor out these code patterns into access helpers and overload them so that all accessors can be coded in a uniform way. This will complicate the encoding, though. • The approach is restricted to single inheritance. 3.3 Functional objects, again with tail polymorphism So far we have defined all methods as separate functions that process “data records”. Hence, we ignored the OO encapsulation premise: our data and methods were divorced from each other. Thereby, we were able to circumvent problems of self references that tend to occur in object encodings. Also, we avoided the classic dichotomy ‘mutable vs. functional objects’. We will complement the picture by exploring a functional object encoding (this section) and a mutable object encoding (next section). We continue to use tail-polymorphic records. In a functional object encoding, object types are necessarily recursive because all mutating methods are modelled as record components that return “self”. In fact, the fundamental, type-theoretic technique is to use equi-recursive types (Pierce &

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Turner, 1994). We must use iso-recursive types instead since Haskell lacks equirecursive types. Extensible shapes are modelled through the following recursive datatype: data Shape w = Shape { getX , getY , setX , setY , moveTo , rMoveTo , draw , shapeTail }

:: :: :: :: :: :: :: ::

Int Int Int -> Int -> Int -> Int -> IO () w

Shape w Shape w Int -> Shape w Int -> Shape w

This type reflects the complete interface of shapes, including getters, setters, and more complex methods. The object constructor is likewise recursive. Recall that recursion models functional mutation, i.e., the construction of a changed object: shape x y d t = Shape { getX , getY , setX , setY , moveTo , rMoveTo , draw , shapeTail }

= = = = = = = =

x y \x’ -> shape x’ y d t \y’ -> shape x y’ d t \x’ y’ -> shape x’ y’ d t \deltax deltay -> shape (x+deltax) (y+deltay) d t d x y t

As before, subtypes are modelled as instantiations of the base-type record. That is, the Rectangle record type is an instance of the Shape record type, where instantiation fixes the type shapeTail somewhat: type Rectangle w = Shape (RectangleDelta w) data RectangleDelta w = RectangleDelta { getWidth’ , getHeight’ , setWidth’ , setHeight’ , rectangleTail }

:: :: :: :: ::

Int Int Int -> Rectangle w Int -> Rectangle w w

We used primed labels because we wanted to save the unprimed names for the actual programmer API. The following implementations of the unprimed functions hide the fact that rectangle records are nested. getWidth getHeight setWidth setHeight

= = = =

getWidth’ getHeight’ setWidth’ setHeight’

. . . .

shapeTail shapeTail shapeTail shapeTail

The constructor for rectangles elaborates the constructor for shapes as follows:

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rectangle x y w h = shape x y drawRectangle shapeTail where drawRectangle x y = putStrLn ("Drawing a Rectangle at:(" ++ (show x) ++ "," ++ (show y) ++ "), width " ++ (show w) ++ ", height " ++ (show h) ) shapeTail = RectangleDelta { getWidth’ = w , getHeight’ = h , setWidth’ = \w’ -> rectangle x y w’ h , setHeight’ = \h’ -> rectangle x y w h’ , rectangleTail = () }

The encoding of the subclass Circle can be derived likewise. (Omitted.) This time, the scribble loop is set up as follows: main = do let scribble = [ narrowToShape (rectangle 10 20 5 6) , narrowToShape (circle 15 25 8) ] mapM_ ( \x -> do draw x draw (rMoveTo x 100 100) ) scribble

The interesting aspect of this encoding concerns the construction of the scribble list. We cast or narrow the shapes of different kinds to a common type. This is a general option, which we could have explored in the previous section (where we used embedding into a union type instead). Narrowing takes a shape with an arbitrary tail and returns a shape with tail (): narrowToShape :: Shape w -> Shape narrowToShape s = s { setX = , setY = , moveTo = , rMoveTo = , shapeTail = }

() narrowToShape . setX s narrowToShape . setY s \z -> narrowToShape . moveTo s z \z -> narrowToShape . rMoveTo s z ()

Assessment of the encoding • The encoding is faithful to the encapsulation premise of the OO paradigm. • The specific extensibility problem of the ‘union type’ approach is resolved (cf. assessment Sec. 3.2). Code that accesses a subtype-polymorphic collection does not need to be revised when new subtypes are added elsewhere in the program. The ‘narrowing’ approach frees the programmer from commitment to a specific union type.

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O. Kiselyov and R. L¨ ammel • The narrowing approach (unlike the union-type one) does not permit downcasting. • The implementation of the narrowing operation is base-type-specific just as the earlier embedding helpers for union types. Boilerplate code of that kind is, of course, not required from programmers in mainstream OO languages. 3.4 Mutable objects, again with tail polymorphism

We also review an object encoding for mutable objects, where we employ IORefs of the IO monad to enable object state — as is the case for OOHaskell. The functions (“methods”) in a record manipulate the state through IORef operations. We continue to use tail-polymorphic records. Extensible shapes are modelled through the following type: data Shape w = Shape { getX , getY , setX , setY , moveTo , rMoveTo , draw , shapeTail }

:: :: :: :: :: :: :: ::

IO Int IO Int Int -> Int -> Int -> Int -> IO () w

IO () IO () Int -> IO () Int -> IO ()

The result type of all methods is wrapped in the IO monad so that all methods may have side effects, if necessary. One may wonder whether this is really necessary for getters. Even for a getter, we may want to add memoisation or logging, when we override the method in a subclass, in which case a non-monadic result type would be too restrictive. The object generator (or constructor) for shapes is parameterised in the initial shape position x and y, in a concrete implementation of the abstract method draw, in the tail of the record to be contributed by the subtype, and in self so to enable open recursion. The latter lets subtypes override method defined in shape. (We will illustrate overriding shortly.) shape x y concreteDraw tail self = do xRef <- newIORef x yRef <- newIORef y tail’ <- tail returnIO Shape { getX = readIORef xRef , getY = readIORef yRef , setX = \x’ -> writeIORef xRef x’ , setY = \y’ -> writeIORef yRef y’ , moveTo = \x’ y’ -> do { setX self x’; setY self y’ } , rMoveTo = \deltax deltay -> do x <- getX self y <- getY self

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moveTo self (x+deltax) (y+deltay) , draw = concreteDraw self , shapeTail = tail’ self }

The type declarations for rectangles are the following: type Rectangle w = Shape (RectangleDelta w) data RectangleDelta w = RectangleDelta { getWidth’ , getHeight’ , setWidth’ , setHeight’ , rectangleTail }

:: :: :: :: ::

IO Int IO Int Int -> IO () Int -> IO () w

Again, we define unprimed names to hide the nested status of the rectangle API: getWidth getHeight setWidth setHeight

= = = =

getWidth’ getHeight’ setWidth’ setHeight’

. . . .

shapeTail shapeTail shapeTail shapeTail

We reveal the object generator for rectangles step by step. rectangle x y w h = shape x y drawRectangle shapeTail where -- to be cont’d

We invoke the generator shape, passing on the normal constructor arguments x and y, a rectangle-specific draw method, and the tail for the rectangle API. We do not yet fix the self reference, thereby allowing for further subtyping of rectangle. We define the draw method as follows, resorting to C++-like syntax, <<, for daisy chaining output: drawRectangle self = putStr "Drawing a Rectangle at:(" << getX self << ls "," << getY self << ls "), width " << getWidth self << ls ", height " << getHeight self << ls "\n"

Finally, the following is the rectangle part of the shape object: shapeTail = do wRef <- newIORef w hRef <- newIORef h returnIO ( \self -> RectangleDelta { getWidth’ , getHeight’ , setWidth’ , setHeight’ , rectangleTail } )

= = = = =

readIORef wRef readIORef hRef \w’ -> writeIORef wRef w’ \h’ -> writeIORef hRef h’ ()

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The overall subtype derivation scheme is at ease with overriding methods in subtypes. We illustrate this capability by temporarily assuming that the draw method is not abstract. So we may revise the constructor for shapes as follows: shape x y tail self = do xRef <- newIORef x yRef <- newIORef y tail’ <- tail returnIO Shape { -- ... as before but we deviate for draw ... , draw = putStrLn "Nothing to draw" }

We override draw when constructing rectangles: rectangle x y w h self = do super <- shape x y shapeTail self returnIO super { draw = drawRectangle self }

As in the previous section, we use narrowToShape when building a list of different shapes. Actual object construction ties the recursive knot for the self references with mfix. Hence, mfix is our operator “new”. main = do s1 <- mfix $ rectangle 10 20 5 6 s2 <- mfix $ circle 15 25 8 let scribble = [ narrowToShape s1 , narrowToShape s2 ] mapM_ ( \x -> do draw x rMoveTo x 100 100 draw x ) scribble

The narrow operation is trivial this time: narrowToShape :: Shape w -> Shape () narrowToShape s = s { shapeTail = () }

We just “chop off” the tail of shape objects. We may no longer use any rectangleor circle-specific methods. One may say that chopping off the tail makes the fields in the tail and the corresponding methods private. The openly recursive methods, in particular draw, had access to self that characterised the whole object, before the chop off. The narrow operation becomes (potentially much) more involved or infeasible once we consider self-returning methods, binary methods, co- and contravariance, and other advanced OO idioms.

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Assessment of the encoding This encoding is actually very close to OOHaskell except that the former uses explicitly declared, non-extensible record types. As a result, the encoding requires substantial boilerplate code (to account for type extension) and subtyping is explicit. Furthermore, OOHaskell leverages typelevel programming to lift restrictions like the limited narrowing capabilities. 3.5 Subtypes as composed record types with overloading Many problems of tail-polymorphic record types prompt us to consider an alternative. We now compose record types for subtypes and use type classes to represent the actual subtype relationships. Such use of type classes has first been presented in (Shields & Peyton Jones, 2001) for encoding of OO interface polymorphism in Haskell. We generalise this technique for class inheritance. The compositional approach can be described as follows: • • • • • •

The data part of an OO class amounts to a record type. Each such record type includes components for superclass data The interface for each OO class amounts to a Haskell type class. OO superclasses are mapped to Haskell superclass constraints. Reusable OO method implementations are mapped to default methods. A subtype is implemented as a type-class instance.

We begin with the record type for the data part of the Shape class: data ShapeData = ShapeData { valX :: Int , valY :: Int }

For convenience, we also provide a constructor: shape x y = ShapeData { valX = x , valY = y }

We define a type class Shape that models the OO interface for shapes: class Shape s where getX :: s -> Int setX :: Int -> s -> s getY :: s -> Int setY :: Int -> s -> s moveTo :: Int -> Int -> s -> s rMoveTo :: Int -> Int -> s -> s draw :: s -> IO () -- to be cont’d

We would like to provide reusable definitions for most of these methods (except for draw of course). In fact, we would like to define the accessors to shape data once and for all. To this end, we need additional helper methods. While it is clear how to define accessors on ShapeData, we must provide generic definitions that are able to handle records that include ShapeData as one of their (immediate or non-immediate) components. This leads to the following two helpers:

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class Shape s where -- cont’d from earlier readShape :: (ShapeData -> t) -> s -> t writeShape :: (ShapeData -> ShapeData) -> s -> s

which let us define generic shape accessors: class Shape s where -- cont’d from earlier getX = readShape valX setX i = writeShape (\s -> s { valX = i }) getY = readShape valY setY i = writeShape (\s -> s { valY = i }) moveTo x y = setY y . setX x rMoveTo deltax deltay s = moveTo x y s where x = getX s + deltax y = getY s + deltay

We do not define an instance of the Shape class for ShapeData because the original shape class was abstract due to the purely virtual draw method. As we move to rectangles, we define their data part as follows: data RectangleData = RectangleData { valShape :: ShapeData , valWidth :: Int , valHeight :: Int }

The rectangle constructor also invokes the shape constructor: rectangle x y w h = RectangleData { valShape = shape x y , valWidth = w , valHeight = h }

“A rectangle is a shape.” We provide access to the shape part as follows: instance Shape RectangleData where readShape f = f . valShape writeShape f s = s { valShape = readShape f s } -- to be cont’d

We also implement the draw method. -- instance Shape RectangleData cont’d draw s = putStrLn ("Drawing a Rectangle at:(" ++ (show (getX s)) ++ "," ++ (show (getY s)) ++ "), width " ++ (show (getWidth s)) ++ ", height " ++ (show (getHeight s)))

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We also need to define a Haskell type class for the OO class of rectangles. OO subclassing coincides in Haskell type-class subclassing. class Shape s => Rectangle s where -- to be cont’d

The type class is derived from the corresponding OO class just as we explained for the base class of shapes. The class defines the ‘normal’ interface of rectangles and access helpers. -- class Rectangle cont’d getWidth :: s -> Int getWidth = readRectangle valWidth setWidth :: Int -> s -> s setWidth i = writeRectangle (\s -> s getHeight :: s -> Int getHeight = readRectangle valHeight setHeight :: Int -> s -> s setHeight i = writeRectangle (\s -> s

{ valWidth = i })

{ valHeight = i })

readRectangle :: (RectangleData -> t) -> s -> t writeRectangle :: (RectangleData -> RectangleData) -> s -> s

“A rectangle is (nothing but) a rectangle.” instance Rectangle RectangleData where readRectangle = id writeRectangle = id

The subclass for circles can be encoded in the same way. The scribble loop can be performed on tagged rectangles and circles: main = do let scribble = [ Left (rectangle 10 20 5 6) , Right (circle 15 25 8) ] mapM_ ( \x -> do draw x draw (rMoveTo 100 100 x) ) scribble

We attach Left and Right tags at the top-level this time. Such simple tagging was not possible with the tail-polymorphic encodings. We still need an instance for Shape that covers tagged shapes: instance (Shape a, Shape b) => Shape (Either a b) where readShape f = either (readShape f) (readShape f) writeShape f = bimap (writeShape f) (writeShape f) draw = either draw draw

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The bi-functorial map, bimap, pushes writeShape into the tagged values. The Either-specific fold operation, either, pushes readShape and draw into the tagged values. For completeness, we recall the relevant facts about bi-functors and folds on Either: class BiFunctor f where bimap :: (a -> b) -> (c -> d) -> f a c -> f b d instance BiFunctor Either where bimap f g (Left x) = Left (f x) bimap f g (Right x’) = Right (g x’) either :: (a -> c) -> (b -> c) -> Either a b -> c either f g (Left x) = f x either f g (Right y) = g y

We should mention a minor but useful variation, which avoids the explicit attachment of tags when inserting into a subtype-polymorphic collection. We use a special cons operation, consEither, which replaces the normal list constructor (:): -- ... so far ... let scribble = [ Left (rectangle 10 20 5 6) , Right (circle 15 25 8) ] -- ... liberalised notation ... let scribble = consEither (rectangle 10 20 5 6) [circle 15 25 8] -- A union-constructing cons operation consEither :: h -> [t] -> [Either h t] consEither h t@(_:_) = Left h : map Right t consEither _ _ = error "Cannot cons with empty tail!"

Assessment of the encoding • This approach is highly systematic and general. For instance, multiple inheritance is immediately possible. One may argue that this approach does not directly encode OO class inheritance. Rather, it mimics object composition. One might indeed convert native OO programs, prior to encoding, so that they do not use class inheritance, but they use interface polymorphism combined with (manually coded) object composition instead. • A fair amount of boilerplate code is required (cf. readShape and writeShape). Also, each transitive subtype relationship requires surprising boilerplate. For example, let us assume an OO class FooBar that is a subclass of Rectangle. The transcription to Haskell would involve three instances: one for the type class that is dedicated to FooBar (“Ok”), one for Rectangle (still “Ok” except the scattering of implementation), and one for Shape (“annoying”).

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• The union-type technique improved compared to Sec. 3.2. The top-level tagging scheme eliminates the need for tagging helpers that are specific to the object types. Also, the consEither operation relieves us from the chore of explicitly writing sequences of tags. However, we must assume that we insert into a non-empty list, and we also must accept that the union type increases for each new element in the list — no matter how many different element types are encountered. Also, if we want to downcast from the union type, we still need to know its exact layout. To lift these restrictions, we have to engage into proper type-class-based programming. 3.6 Variation — existential quantification So far we have restricted ourselves to Haskell 98. We now turn to common extensions of Haskell 98, in an attempt to improve on the problems that we have encountered. In upshot, we cannot spot obvious ways for improvement. Our first attempt is to leverage existential quantification for the implementation of subtype-polymorphic collections. Compared to the earlier Either-based approach, we homogenise shapes by making them opaque (Cardelli & Wegner, 1985) as opposed to embedding them into the union type. This use of existentials could be combined with various object encodings; we illustrate it here for the specific encoding from the previous section. We define an existential envelope for shape data. data OpaqueShape = forall x. Shape x => HideShape x

“Opaque shapes are (still) shapes.” Hence, a Shape instance: instance Shape where readShape f writeShape f draw

OpaqueShape (HideShape x) = readShape f x (HideShape x) = HideShape $ writeShape f x (HideShape x) = draw x

When building the scribble list, we place shapes in the envelope. let scribble = [ HideShape (rectangle 10 20 5 6) , HideShape (circle 15 25 8) ]

Assessment of the encoding • Compared to the ‘union type’ approach, programmers do not have to invent union types each time they need to homogenise different subtypes. Instead, all shapes are tagged by HideShape. The ‘narrowing’ approach was quite similar, but it required boilerplate. • We face a new problem. Existential quantification limits type inference. We see that in the definition of OpaqueShape; viz. the explicit constraint Shape. It is mandatory to constraint the quantifier by all subtypes whose methods may be invoked. A reader may notice the similar problem for the ‘union type’ approach, which required Shape constraints in the instance

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instance (Shape a, Shape b) => Shape (Either a b) where ...

That instance, however, was merely a convenience. We could have disposed of it and used the fold operation either explicitly in the scribble loop: main = do let scribble = [ Left (rectangle 10 20 5 6) , Right (circle 15 25 8) ] mapM_ (either scribbleBody scribbleBody) scribble where scribbleBody x = do draw x draw (rMoveTo 100 100 x)

By contrast, the explicit constraint for the existential envelope cannot be eliminated. Admittedly, the loss of type inference is a nuance in this specific example. In general, however, this weakness of existentials is quite annoying. It is intellectually dissatisfying since type inference is one of the added values of an (extended) Hindley/Milner type system, when compared to mainstream OO languages. Worse than that, the kind of constraints in the example are not necessary in mainstream OO languages (without type inference), because these constraints deal with subtyping, which is normally implicit. We do not use existentials in OOHaskell. 3.7 Variation — heterogeneous collections We continue with our exploration of common extensions of Haskell 98. In fact, we will offer another option for the difficult problem of subtype-polymorphic collections. We recall that all previously discussed techniques aimed at making it possible to construct a normal homogeneous Haskell list in the end. This time, we will engage into the construction of a heterogeneous collection in the first place. To this end, we leverage techniques as described by us in the HList paper (Kiselyov et al., 2004). Heterogeneous collections rely on multi-parameter classes (Chen et al., 1992; Jones, 1992; Jones, 1995; Peyton Jones et al., 1997) with functional dependencies (Jones, 2000; Duck et al., 2004). Heterogeneous lists are constructed with dedicated constructors HCons and HNil — analogues of (:) and []. One may think of a heterogeneous list type as a nested binary product, where HCons corresponds to (,) and HNil to (). We use special HList functions to process the heterogeneous lists; the example requires a map operation. The scribble loop is now encoded as follows: main = do let scribble =

HCons (rectangle 10 20 5 6) (HCons (circle 15 25 8) HNil) hMapM_ (undefined::ScribbleBody) scribble

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The operation hMapM_ is the heterogeneous variation on the normal monadic map mapM_. The function argument for the map cannot be given inline; instead we pass a proxy undefined::ScribbleBody. This detour is necessary due to technical reasons that are related to the combination of rank-2 polymorphism and type-class-bounded polymorphism.6 The type code for the body of the scribble loop is defined by a trivial datatype: data ScribbleBody -- No constructors needed; non-Haskell 98

The heterogeneous map function is constrained by the Apply class, which models interpretation of function codes like ScribbleBody. Here is the Apply class and the instance dedicated to ScribbleBody: class Apply f a r | f a -> r where apply :: f -> a -> r instance Shape s => Apply ScribbleBody s (IO ()) where apply _ x = do draw x draw (rMoveTo 100 100 x)

Assessment of the encoding • This approach eliminates all effort for inserting elements into a collection. • The approach comes with heavy surface encoding; cf. type code ScribbleBody. • This encoding is at odds with type inference — just as in the case of existentials. That is, the Apply instance must be explicitly constrained by the interfaces that are going to be relied upon in the body of the scribble loop. Again, the amount of explicit typing is not yet disturbing in the example at hand, but it is an intrinsic weakness of the encoding. The sort of required explicit typing goes beyond standard OO programming practise. 4 Type-agnostic OOHaskell idioms We will now systematically develop all important OOHaskell programming idioms. In this section, we will restrict ourselves to ‘type-agnostic’ idioms, as to clearly substantiate that most OOHaskell programming does not require type declarations, type annotations, explicit casts for object types — thanks to Haskell’s type inference and its strong support for polymorphism. The remaining, ‘type-perceptive’ 6

A heterogeneous map function can encounter entities of different types. Hence, its argument function must be polymorphic on its own (which is different from the normal map function for lists). The argument function typically uses type-class-bounded polymorphic functions to process the entities of different types. The trouble is that the map function cannot possibly anticipate all the constraints required by the different uses of the map function. The type-code technique moves the constraints from the type of the heterogeneous map function to the interpretation site of the type codes.

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OOHaskell idioms (including a few advanced topics related to subtyping) are described in the subsequent section. In both sections, we adopt the following style. We illustrate the OO idioms and describe the technicalities of encoding. We highlight strengths of OOHaskell: support for the traditional OO idioms as well as extra features due to the underlying record calculus, and first-class status of labels, methods and classes. Finally, we illustrate the overall programmability of a typed OO language design in Haskell. As a matter of style, we somewhat align the presentation of OOHaskell with the OCaml object tutorial. Among the many OO systems that are based on open records (Perl, Python, Javascript, Lua, etc.), OCaml stands out because it is statically typed (just as OOHaskell). Also, OCaml (to be precise, its predecessor ML-ART) is close to OOHaskell in terms of motivation: both aim at the introduction of objects as a library in a strongly-typed functional language with type inference. The implementation of the libraries and the sets of features used or required are quite different (cf. Sec. 6.2.1 for a related work discussion), which makes a comparison even more interesting. Hence, we draw examples from the OCaml object tutorial, to specifically contrast OCaml and OOHaskell code and to demonstrate the fact that OCaml examples are expressible in OOHaskell, roughly in the same syntax, based on direct, local translation. We also use the OCaml object tutorial because it is clear, comprehensive and concise.

4.1 Objects as records Quoting from (Leroy et al. , 2004)[§ 3.2]:7 “The class point below defines one instance variable varX and two methods getX and moveX. The initial value of the instance variable is 0. The variable varX is declared mutable. Hence, the method moveX can change its value.” class point = object val mutable varX = 0 method getX = varX method moveX d = varX <- varX + d end;;

4.1.1 First-class labels The transcription to OOHaskell starts with the declaration of all the labels that occur in the OCaml code. The HList library readily offers 4 different models of labels. In all cases, labels are Haskell values that are distinguished by their Haskell type. We choose the following model: • The value of a label is “⊥”. 7

While quoting portions of the OCaml tutorial, we take the liberty to rename some identifiers and to massage some subminor details.

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• The type of a label is a proxy for an empty type (empty except for “⊥”).8 data VarX; varX = proxy :: Proxy VarX data GetX; getX = proxy :: Proxy GetX data MoveX; moveX = proxy :: Proxy MoveX

where proxies are defined as data Proxy e proxy :: Proxy e proxy = ⊥

-- A proxy type is an empty phantom type. -- A proxy value is just ‘‘⊥’’.

Simple syntactic sugar can significantly reduce the length of the one-liners for label declaration should this become an issue. For instance, we may think of the above lines just as follows: -- Syntax extension assumed; label is a new keyword. label varX label getX label moveX

The explicit declaration of OOHaskell labels blends well with Haskell’s scoping rules and its module concept. Labels can be private to a module, or they can be exported, imported, and shared. All models of HList labels support labels as firstclass citizens. In particular, we can pass them to functions. The “labels as type proxies” idea is the basis for defining record operations since we can thereby dispatch on labels in type-level functionality. We will get back to the record operations shortly. 4.1.2 Mutable variables The OCaml point class is transcribed to OOHaskell as follows: point = do x <- newIORef 0 returnIO $ varX .=. x .*. getX .=. readIORef x .*. moveX .=. (\d -> do modifyIORef x (+d)) .*. emptyRecord

The OOHaskell code clearly mimics the OCaml code. While we use Haskell’s IORefs to model mutable variables, we do not use any magic of the IO monad. We could as well use the simpler ST monad, which is very well formalised (Launchbury & Peyton Jones, 1995). The source distribution for the paper illustrates the ST option. 8

It is a specific GHC extension of Haskell 98 to allow for datatypes without any constructor declarations. Clearly, this is a minor issue because one could always declare a dummy constructor that is not used by the program.

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The Haskell representation of the point class stands revealed as a value binding declaration of a monadic record type. The do sequence first creates an IORef for the mutable variable, and then returns a record for the new point object. In general, the OOHaskell records provide access to the public methods of an object and to the IORefs for public mutable variables. We will often call all record components of OOHaskell’s objects just ‘methods’. In the example, varX is public, just as in the original OCaml code. In OOHaskell, a private variable would be encoded as an IORef that is not made available through a record component. (Private variables were explored in the shapes example.) 4.1.3 HList records We may ask Haskell to tell us the inferred type of point: ghci> :t point point :: IO (Record (HCons (Proxy MutableX, IORef Integer) (HCons (Proxy GetX, IO Integer) (HCons (Proxy Move, Integer -> IO ()) HNil))))

The type reveals the use of HList’s extensible records (Kiselyov et al., 2004). We explain some details about HList, as to make the present paper self-contained. The inferred type shows that records are represented as heterogeneous label-value pairs, which are promoted to a proper record type through the type-constructor Record. -- HList constructors data HNil = HNil -- empty heterogeneous list data HCons e l = HCons e l -- non-empty heterogeneous list -- Sugar for forming label-value pairs infixr 4 .=. l .=. v = (l,v) -- Record type constructor newtype Record r = Record r

The constructor Record is opaque for the library user. Instead, the library user (and most of the library code itself) relies upon a constrained constructor: -- Record value constructor mkRecord :: HRLabelSet r => r -> Record r mkRecord = Record

The constraint HRLabelSet r statically assures that all labels are pairwise distinct as this is a necessary precondition for a list of label-value pairs to qualify as a record. (We omit the routine specification of HRLabelSet r (Kiselyov et al., 2004).) We can now implement emptyRecord, which was used in the definition of point: emptyRecord = mkRecord HNil

The record extension operator, (.*.), is a constrained variation on the heterogeneous cons operation, HCons: we need to make sure that the newly added label-value pair does not violate the uniqueness property for the labels. This is readily expressed by wrapping the unconstrained cons term in the constrained record constructor:

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infixr 2 .*. (l,v) .*. (Record r) = mkRecord (HCons (l,v) r)

4.1.4 OO test cases We want to instantiate the point class and invoke some methods. We begin with an OCaml session, which shows some inputs and the responses from the OCaml interpreter:9 # let p = new point;; val p : point = # p#getX;; - : int = 0 # p#moveX 3;; - : unit = () # p#getX;; - : int = 3

In Haskell, we capture this program in a monadic do sequence because method invocations can involve IO effects including the mutation of objects. We denote method invocation by (#), just as in OCaml; this operation is a plain record lookup. Hence: myFirstOOP = do p <- point -- no need for new! p # getX >>= Prelude.print p # moveX $ 3 p # getX >>= Prelude.print

OOHaskell and OCaml agree: ghci> myFirstOOP 0 3

For completeness we outline the definition of “#”: -- Sugar operator infixr 9 # obj # feature = hLookupByLabel feature obj -- Type-level operation for look-up class HasField l r v | l r -> v where hLookupByLabel:: l -> r -> v

This operation performs type-level (and value-level) traversal of the label-value pairs, looking up the value component for a given label from the given record, while 9

OCaml’s prompt is indicated by a leading character “#”. Method invocation is modelled by the infix operator “#”. The lines with leading “val” or “–” are the responses from the interpreter.

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using the label type as a key. We recall that the term ‘field’ (cf. HasField) originates from record terminology. In OOHaskell, all ‘fields’ are ‘methods’. (We omit the routine specification of HasField l r v (Kiselyov et al., 2004).) The class declaration reveals that HList (and thereby OOHaskell) relies on multi-parameter classes (Chen et al., 1992; Jones, 1992; Jones, 1995; Peyton Jones et al., 1997) with functional dependencies (Jones, 2000; Duck et al., 2004). 4.2 Object generators In class-based, mainstream OO languages, the construction of new class instances is normally regulated by so-called constructor methods. In OOHaskell, instances are created by a function that serves as an object generator. The function can be seen as the embodiment of the class itself. The point computation defined above is a trivial example of an object generator. 4.2.1 Constructor arguments Quoting from (Leroy et al. , 2004)[§ 3.1]: “The class point can also be abstracted over the initial value of varX. The parameter x_init is, of course, visible in the whole body of the definition, including methods. For instance, the method getOffset in the class below returns the position of the object relative to its initial position.” class para_point x_init = object val mutable varX = x_init method getX = varX method getOffset = varX - x_init method moveX d = varX <- varX + d end;;

In OOHaskell, objects are created as the result of monadic computations producing records. We can parameterise these computations by turning them into functions, object generators, which take construction parameters as arguments. For instance, the parameter x_init of the OCaml class para_point ends up as a plain function argument: para_point x_init = do x <- newIORef x_init returnIO $ varX .=. x .*. getX .=. readIORef x .*. getOffset .=. queryIORef x (\v -> v - x_init) .*. moveX .=. (\d -> modifyIORef x (+d)) .*. emptyRecord

4.2.2 Construction-time computations Quoting from (Leroy et al. , 2004)[§ 3.1]:

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“Expressions can be evaluated and bound before defining the object body of the class. This is useful to enforce invariants. For instance, points can be automatically adjusted to the nearest point on a grid, as follows:” class adjusted_point x_init = let origin = (x_init / 10) * 10 in object val mutable varX = origin method getX = varX method getOffset = varX - origin method moveX d = varX <- varX + d end;;

In OOHaskell, we follow the suggestion from the OCaml tutorial: we use local let bindings to carry out the constructor computations “prior” to returning the constructed object: adjusted_point x_init = do let origin = (x_init ‘div‘ 10) * 10 x <- newIORef origin returnIO $ varX .=. x .*. getX .=. readIORef x .*. getOffset .=. queryIORef x (\v -> v - origin) .*. moveX .=. (\d -> modifyIORef x (+d)) .*. emptyRecord

That “prior” is not meant in a temporal sense: OOHaskell remains a non-strict language, in contrast to OCaml. 4.2.3 Implicitly polymorphic classes A powerful feature of OOHaskell is implicit polymorphism for classes. For instance, the class para_point is polymorphic with regard to the point’s coordinate — without our contribution. This is a fine difference between the OCaml model and our OOHaskell transcription. In OCaml’s definition of para_point, the parameter x_init was of the type int — because the operation (+) in OCaml can deal with integers only. The OOHaskell points are polymorphic — a point’s coordinate can be any Num-ber, for example, an Int or a Double. Here is an example to illustrate that: myPolyOOP = do p <- para_point (1::Int) p’ <- para_point (1::Double) p # moveX $ 2 p’ # moveX $ 2.5 p # getX >>= Prelude.print p’ # getX >>= Prelude.print

The OOHaskell points are actually bounded polymorphic. The point coordinate may be of any type that implements addition. Until very recently, one could

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not express this in Java and in C#. Expressing bounded polymorphism in C++ is possible with significant contortions. In (OO)Haskell, we did not have to do anything at all. Bounded polymorphism (aka, generics) is available in Ada95, Eiffel and a few other languages. However, in those languages, the polymorphic type and the type bounds must be declared explicitly. (There are ongoing efforts to add some specific bits of type inference to new versions of mainstream OO languages.) In (OO)Haskell, the type system infers the (bounded) polymorphism on its own, in full generality. The implicit polymorphism of OOHaskell does not injure static typing. If we confuse Ints and Doubles in the above code, e.g., by attempting “p # moveX $ 2.5”,then we get a type error saying that Int is not the same as Double. In contrast, the poor men’s implementation of polymorphic collections, e.g., in Java < 1.5, which up-casts an element to the most general Object type when inserting it into the collection, requires runtime-checked downcasts when accessing elements. 4.2.4 Nested object generators Quoting from (Leroy et al. , 2004)[§ 3.1]: “The evaluation of the body of a class only takes place at object creation time. Therefore, in the following example, the instance variable varX is initialised to different values for two different objects.” let x0 = ref 0;; class incrementing_point : object val mutable varX = incr x0; !x0 method getX = varX method moveX d = varX <- varX + d end;;

We test this new class at the OCaml prompt: # # -

new incrementing_point#getX;; : int = 1 new incrementing_point#getX;; : int = 2

The variable x0 can be viewed as a “class variable”, belonging to a class object. Recall that classes are represented by object generators in OOHaskell. Hence to build a class object we need a nested object generator: incrementing_point = do x0 <- newIORef 0 returnIO ( do modifyIORef x0 x <- readIORef returnIO $ varX .=. .*. getX .=.

(+1) x0 >>= newIORef x readIORef x

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.*. moveX .=. (\d -> modifyIORef x (+d)) .*. emptyRecord)

We can nest generators to any depth since we just use normal Haskell scopes. In the example, the outer level does the computation for the point template (i.e., “class”); the inner level constructs points themselves. Here is a more suggestive name for the nested generator: makeIncrementingPointClass = incrementing_point

This (trivial) example demonstrates that classes in OOHaskell are really firstclass citizens. We can pass classes as arguments to functions and return them as results. In the following code fragment, we create a class in a scope, and bind it as a value to a locally-scoped variable, which is then used to instantiate the created class in that scope. The localClass is a closure over the mutable variable x0. myNestedOOP = do localClass <- makeIncrementingPointClass localClass >>= ( # getX ) >>= Prelude.print localClass >>= ( # getX ) >>= Prelude.print ghci> myNestedOOP 1 2

In contrast, such a class closure is not possible in Java, let alone C++. Java supports anonymous objects, but not anonymous first-class classes. Nested classes in Java must be linked to an object of the enclosing class. Named nested classes in C# are free from that linking restriction. However, C# does not support anonymous classes or class computations in a local scope (although anonymous delegates of C# let us emulate computable classes). Nevertheless, classes, as such, are not first-class citizens in any mainstream OO language. 4.2.5 Open recursion The methods of an object may send messages to ‘self’. To support inheritance with override that ‘self’ must be bound explicitly (Cook, 1989). Otherwise, inheritance will not able to revise the messages to self that were coded in a superclass. Consequently, general object generators are to be given in the style of ‘open recursion’: they take self and construct (some part of) self. Quoting from (Leroy et al. , 2004)[§ 3.2]: “A method or an initialiser can send messages to self (that is, the current object). For that, self must be explicitly bound, here to the variable s (s could be any identifier, even though we will often choose the name self.) ... Dynamically, the variable s is bound at the invocation of a method. In particular, when the class printable_point is inherited, the variable s will be correctly bound to the object of the subclass.” class printable_point x_init = object (s) val mutable varX = x_init

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O. Kiselyov and R. L¨ ammel method getX method moveX d method print end;;

= varX = varX <- varX + d = print_int s#getX

Again, this OCaml code is transcribed to OOHaskell very directly. The self argument, s, ends up as an ordinary argument of the monadic function for generating printable point objects: printable_point x_init s = do x <- newIORef x_init returnIO $ varX .=. x .*. getX .=. readIORef x .*. moveX .=. (\d -> modifyIORef x (+d)) .*. print .=. ((s # getX ) >>= Prelude.print) .*. emptyRecord

In OCaml, we use printable_point as follows: # let p = new printable_point 7;; val p : printable_point = # p#moveX 2;; - : unit = () # p#print;; 9- : unit = ()

Although s does not appear on the line that constructs a point p with the new construct, the recursive knot clearly is tied right there. In OOHaskell, we use the (monadic) fixpoint function, mfix, rather than a special keyword new. This makes the nature of openly recursive object generators manifest. mySelfishOOP = do p <- mfix (printable_point 7) p # moveX $ 2 p # print ghci> mySelfishOOP 9

4.2.6 Instantiation checking One potential issue with open recursion in OOHaskell is that some type errors in messages to self will not be spotted until the first object construction is coded. For instance, an OO library developer could, accidentally, provide object generators that turn out to be uninstantiatable; the library user would notice this defect once the generators are put to work. This issue is readily resolved as follows. When we program object generators, we may use the concrete operation: -- An assured printable point generator concrete_printable_point x_init = concrete $ printable_point x_init

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-- The concrete operation concrete generator self = generator self where _ = mfix generator

Operationally, concrete is the identity function. However, it constrains the type of generator such that the application of mfix is typeable. This approach needs to be slightly refined to cover abstract methods (aka pure virtual methods). To this end, one would need to engage into local inheritance — adding vacuous (i.e., potentially undefined) methods for any needed virtual method. This generalised concrete operation would take the virtual portion of a record, or preferably just a proxy for it, so that the purpose of this argument is documented.

4.3 Reuse techniques The first-class status of labels, methods and classes enables various, common and advanced forms of reuse. Single inheritance boils down to (monadic) function composition of object generators. Multiple inheritance and object composition employ more advanced operations of the record calculus.

4.3.1 Single inheritance with extension Quoting from (Leroy et al. , 2004)[§ 3.7]:10 “We illustrate inheritance by defining a class of colored points that inherits from the class of points. This class has all instance variables and all methods of class point, plus a new instance variable color, and a new method getColor.” class colored_point x (color : string) = object inherit point x val color = color method getColor = color end;;

Here is the corresponding OCaml session: # let p’ = new colored_point 5 "red";; val p’ : colored_point = # p’#getX, p’#getColor;; - : int * string = (5, "red")

The following OOHaskell version does not employ a special inherit construct. We compose computations instead. To construct a colored point we instantiate the superclass while maintaining open recursion, and extend the intermediate record, super, by the new method getColor. 10

We use British spelling consistently in this paper, except for some words that enter the text through code samples: color, colored, ...

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colored_point x_init (color::String) self = do super <- printable_point x_init self returnIO $ getColor .=. (returnIO color) .*. super

Here, super is just a variable rather than an extra construct. myColoredOOP = do p’ <- mfix (colored_point 5 "red") x <- p’ # getX c <- p’ # getColor Prelude.print (x,c)

OOHaskell and OCaml agree: ghci> myColoredOOP (5,"red")

4.3.2 Class-polymorphic functionality We can parameterise computations with respect to classes. myFirstClassOOP point_class = do p <- mfix (point_class 7) p # moveX $ 35 p # print ghci> myFirstClassOOP printable_point 42

The function myFirstClassOOP takes a class (i.e., an object generator) as an argument, instantiates the class, and moves and prints the resulting object. We can pass myFirstClassOOP any object generator that creates an object with the slots moveX and print. This constraint is statically verified. For instance, the colored point class, which we derived from the printable point class in the previous section, is suitable: ghci> myFirstClassOOP $ flip colored_point "red" 42

4.3.3 Single inheritance with override We can override methods and still refer to their superclass implementations (akin to the super construct in OCaml and other languages). We illustrate overriding with a subclass of colored_point whose print method is more informative: colored_point’ x_init (color::String) self = do super <- colored_point x_init color self return $ print .=. ( do putStr "so far - "; super # print putStr "color - "; Prelude.print color ) .<. super

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The first step in the monadic do sequence constructs an old-fashioned colored point, and binds it to super for further reference. The second step in the monadic do sequence returns super updated with the new print method. The HList operation “.<.” denotes type-preserving record update as opposed to the familiar record extension “.*.”. The operation “.<.” makes the overriding explicit (as it is in C#, for example). We could also use a hybrid record operation, which does extension in case the given label does not yet occur in the given record, falling back to typepreserving update. This hybrid operation would let us model the implicit overriding in C++ and Java. Again, such a variation point demonstrates the programmability of OOHaskell’s object system. Here is a demo that shows overriding to properly affect the print method: myOverridingOOP = do p <- mfix (colored_point’ 5 "red") p # print ghci> myOverridingOOP so far - 5 color - "red"

4.3.4 Orphan methods We can program methods outside of any hosting class. Such methods can be reused across classes without any inheritance relationship. For instance, we may define a method print_getX that can be shared by all objects that have at least the method getX of the type Show a => IO a — regardless of any inheritance relationships: print_getX self = ((self # getX ) >>= Prelude.print)

We can update the earlier code for printable_point as follows: -- before: inlined definition of print ... .*. print .=. ((s # getX ) >>= Prelude.print) -- after: reusable orphan method ... .*. print .=. print_getX s

4.3.5 Flexible reuse schemes In addition to single class inheritance, there are several other established reuse schemes in OO programming including object composition, different forms of mixins and different forms of multiple inheritance. Given the first-class status of all OOHaskell entities and its foundation in a powerful record calculus, it should be possible to reconstruct most if not all existing reuse schemes. We will use an (admittedly contrived) example to demonstrate a challenging combination of multiple inheritance and object composition. To the best of our knowledge, this example cannot be directly represented in any existing mainstream language. We are going to work through a scenario of making a class heavy_point from three different concrete subclasses of abstract_point. The first two concrete points

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concrete_point1

concrete_point2

concrete_point3

nonshared and closed

shared and open

heavy_point

Fig. 2. A complex reuse scenario will be shared in the resulting heavy point, because we leave open the recursive knot. The third concrete point does not participate in the open recursion and is not shared. In C++ terminology, abstract_point is a virtual base class (with respect to the first two concrete points) and a non-virtual base class at the same time. See Fig. 2 for an overview. The object template for heavy points starts as follows: heavy_point x_init color self = do super1 <- concrete_point1 x_init self super2 <- concrete_point2 x_init self super3 <- mfix (concrete_point3 x_init) ... -- to be continued

We bind all ancestor objects for subsequent reference. We pass self to the first two points, which participate in open recursion, but we fix the third point in place. The first two classes are thus reused in the sense of inheritance, while the third class is reused in the sense of object composition. A heavy point carries print and moveX methods delegating corresponding messages to all three points: ... -- continued from above let myprint = do putStr "super1: "; (super1 putStr "super2: "; (super2 putStr "super3: "; (super3 let mymove = ( \d -> do super1 # moveX $ d super2 # moveX $ d super3 # moveX $ d return $ print .=. myprint .*. moveX .=. mymove .*. emptyRecord ... -- to be continued

# print) # print) # print)

)

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The three points, with all their many fields and methods, contribute to the heavy point by means of left-biased union on records, which is denoted by “.<++.” below: ... -.<++. .<++. .<++.

continued from above super1 super2 super3

Here is a demo: myDiamondOOP = do p <- mfix (heavy_point 42 "blue") p # print -- All points still agree! p # moveX $ 2 p # print -- The third point lacks behind! ghci> myDiamondOOP super1: 42 super2: 42 super3: 42 super1: 46 super2: 46 super3: 44

For comparison, in OCaml, multiple inheritance follows fixed rules. Only the last definition of a method is kept: the redefinition in a subclass of a method that was visible in the parent class overrides the definition in the parent class. Previous definitions of a method can be reused by binding the related ancestor using a special . . . as . . . notation. The bound name is said to be a ‘pseudo value identifier’ that can only be used to invoke an ancestor method. Eiffel, C++, etc. have their own fixed rules and notations for multiple inheritance. OOHaskell allows us to “program” this aspect of the OO type system. Programmers (or language designers) may devise their own inheritance and object composition rules. 4.4 Safe value recursion The support for open recursion in an OO system has a subtle but fundamental difficulty. Of the three ways to emulate open recursion – recursive types, existential abstraction ((R´emy, 1994; Pierce & Turner, 1994)) and value recursion, the latter is the simplest one. This is the one we have chosen for OOHaskell. Recall that each object generator receives the self argument (representing the constructed object), which lets the methods send the messages to the object itself. An object is constructed by obtaining the fixpoint of the generator. Here is a variation on the printable point example from Sec. 4.2.5 that illustrates the potential unsafety of value recursion: printable_point x_init self = do x <- newIORef x_init self # print -- Unsafe!

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O. Kiselyov and R. L¨ ammel returnIO $ varX .=. x .*. getX .=. readIORef x .*. moveX .=. (\d -> modifyIORef x (+d)) .*. print .=. ((self # getX ) >>= Prelude.print) .*. emptyRecord

An object generator may be tempted to invoke methods on the received self argument, as self # print above. That code typechecks. However, the attempt to construct an object by executing mfix (printable_point 0) reveals the problem: looping. Indeed, self represents the object that will be constructed. It is not proper to invoke any method on self when the object generation is still taking place, because self as a whole does not yet exist. In Haskell, accessing a not-yet-constructed object leads to “mere” looping. This, so-called left-recursion problem (well-known in parsing (Aho et al., 1986)) is akin to the divergence of the following trivial expression: mfix (\self -> do { Prelude.print self; return "s" }) In a non-strict language like Haskell, determining the fixpoint of a value a->a where a is not a function type is always safe: the worst can happen is the divergence, but no undefined behaviour. In strict languages, the problem is far more serious: accessing the field before it was filled in is accessing a dummy value (e.g., a null pointer) that was placed into the field prior to the evaluation of the recursive definition. Such an access results in undefined behaviour, and has to be prevented with a run-time check. As noted in (R´emy, 1994), this problem has been widely discussed but no satisfactory solution was found. Although the problem is relatively benign in OOHaskell and never leads to undefined behaviour, we would like to statically prevent it. To be precise, we impose the rule that the constructor may not execute any actions that involve notyet-constructed objects. With little changes, we statically enforce that restriction. Object construction may be regarded as a sort of a staged computation; the problem of preventing the use of not-yet-constructed values is one of the key challenges in multi-staged programming (Taha & Nielsen, 2003), where it has been recently solved with environment classifiers. Our solution is related in principle (making the stage of completion of an object a part of its type), but differs in technique (we exploit compile-time tagging and monadic types rather than higher-ranked types). We introduce a tag NotFixed to mark the objects that are not constructed yet: newtype NotFixed a = NotFixed a -- data constructor opaque!

Because NotFixed is a newtype, this tag imposes no run-time overhead. We do not export the data constructor NotFixed so the user may not arbitrarily introduce or remove that tag. All operations on this tag are restricted to a new module NotFixed that is part of the OOHaskell library. The module exports two new operations: new and construct. The former is a variant of mfix for the IO monad. The construct operation is a variation on returnIO.

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new :: (NotFixed (Record a) -> IO (NotFixed (Record a))) -- object generator -> IO (Record a) -- object computation new f = mfix f >>= (\(NotFixed a) -> return a) construct :: NotFixed (Record a) -> (Record a -> Record b) -> IO (NotFixed (Record b)) construct (NotFixed self) f = returnIO

-- self -- constructor -- object computation $ NotFixed (f self)

Staged object construction proceeds as follows. The argument self passed to the object generator is marked as NotFixed. After the fixpoint is computed, new removes the NotFixed tag. The function construct, while maintaining the NotFixed tag, lifts the tag internally so that the methods being defined by the object generator could use the self reference. We can now write our example as follows: printable_point x_init self = do x <- newIORef x_init -- self # print construct self (\self-> mutableX .=. x .*. getX .=. readIORef x .*. moveX .=. (\d -> modifyIORef x ((+) d)) .*. print .=. ((self # getX ) >>= Prelude.print) .*. emptyRecord) test_pp do p p p

= <- new (printable_point 7) # moveX $ 2 # print

If we uncomment the statement self # print we will get the type error saying that a NotFixed object does not have the method print. (There are no HasField instances for the NotFixed type.) Within the body of construct, the reference to self is available without the NotFixed tag; so one may be tempted to invoke methods on self and execute their actions. However, the second argument of construct is a non-monadic function of the type Record a -> Record b. Because the result type of the function does not include IO, it is not possible to read and write IORef and do other IO actions within that function. In Haskell (in contrast to OCaml), imperativeness of a function is manifest in its type. The extension to the construction of inherited classes is straightforward. For example, the colored_point example from Sec. 4.3.1 now reads: colored_point x_init (color::String) self = do p <- printable_point x_init self -- p # print -- would not typecheck. construct p $ \p -> getColor .=. (returnIO color) .*. p myColoredOOP = do

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O. Kiselyov and R. L¨ ammel p’ <- new (colored_point 5 "red") x <- p’ # getX c <- p’ # getColor Prelude.print (x,c)

The constructor colored_point receives the argument self marked as not-yetconstructed. We pass that argument to printable_point, which gives us a notyet-constructed object of the superclass. We cannot execute any methods on that object (and indeed, uncommenting the statement p # print leads to a type error). The execution of a superclass method may involve the invocation of a method on self (as is the case for the method print), and self is not constructed yet. The construct operation shown here is not fully general; the source distribution illustrates safe object generation where methods refer both to self and super. The technique readily generalises to multiple inheritance and object composition.

5 Type-perceptive OOHaskell idioms So far we have avoided type declarations, type annotations and explicit coercions of object types. We will now discuss those OOHaskell programming scenarios that can benefit from additional type information, or even require it. We will pay special attention to various subtyping and related cast and variance issues. In particular, we will cover the technical details of subtype-polymorphic collections, which require some amount of type perceptiveness, as we saw in the introductory shapes example in Sec. 2.

5.1 Semi-implicit upcasting There is an important difference between OOHaskell’s subtype polymorphism (which we share to some extent with OCaml and ML-ART) and polymorphism in C++ and other mainstream OO languages.11 In the latter languages, an object can be implicitly coerced to an object of any of its superclasses (“upcast”). One may even think that an object is polymorphic by itself, i.e., it has types of all of its superclasses, simultaneously. Hence, there is no need for functions on objects (or methods) to be polymorphic by themselves; they are monomorphic. In OCaml and OOHaskell, it is the other way around: objects are monomorphic (with regard to the record structure) and the language semantics does not offer any implicit upcasting or narrowing.12 However, functions that take objects can be polymorphic and can process objects of different types. To be precise, OOHaskell exploits type-class-bounded polymorphism. A function that takes an object and refers to its methods (i.e., record components) has in its inferred or explicit type HasField constraints for these record components. The function therefore accepts

11 12

See, however, Sec. 5.7, where we emulate the mainstream nominal subtyping. We prefer the term narrow over up-cast, as to emphasise the act of restricting the interface of an object, as opposed to walking up an explicit (perhaps even nominal) subtyping hierarchy.

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any object that has at least the components that satisfy the HasField constraints.13 Therefore, most of the time no (implicit or explicit) upcasting is needed; in fact, in Sec. 4 we did not see any. An explicit cast is usually understood as casting to an explicitly named target type. We discuss such casts later in this section. Here we show that the established explicit-vs.-implicit upcast dichotomy misses an intermediate option, which is admitted by OOHaskell. Namely, OOHaskell lets the programmer specify that narrowing is to be performed — without giving an explicit target type, though. So we continue to get by without specifying types, leaving it all to type inference — at least for a while. In OOHaskell (and OCaml), we must narrow an object if its expression context has no type-class-bounded polymorphism left and requires an object of a different type. The archetypal example is placing objects in a homogeneous collection, e.g., a list. The original item objects may be of different types; therefore, we must establish a common element type and narrow the items to it. This common element type does not have to be specified explicitly, however. OOHaskell can compute that common type as we add objects to the collection; the context will drive the narrowing. The OOHaskell implementation of the shapes example in Sec. 2 involved this sort of narrowing: myOOP = do s1 <- mfix (rectangle (10::Int) (20::Int) 5 6) s2 <- mfix (circle (15::Int) 25 8) let scribble = consLub s1 (consLub s2 nilLub) ... and so on ...

The designated list constructors nilLub and consLub incorporate narrowing into their normal constructor behaviour. The specific element type of each new element constraints the ultimate least-upper bound (LUB) element type for the final list. Elements are continuously cast towards this LUB. The list constructors are defined as follows: -- A type-level code for the empty list data NilLub -- The empty list constructor nilLub = ⊥ :: NilLub -- Cons as a type-level function class ConsLub h t l | h t -> l where consLub :: h -> t -> l -- No coercion needed for a singleton list instance ConsLub e NilLub [e] where consLub h _ = [h] 13

We oversimplify here by not talking about operations that add or remove fields. This is a fair simplification though because we talk about normal OO functionality here, as opposed to free-wheeling functionality for record manipulation.

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-- Narrow head and tail to their LUB type instance LubNarrow e0 e1 e2 => ConsLub e0 [e1] [e2] where consLub h t = fst (head z) : map snd (tail z) where z = map (lubNarrow h) (⊥:t)

The important operation is lubNarrow: class LubNarrow a b c | a b -> c where lubNarrow :: a -> b -> (c,c)

Given two values of record types a and b, this operation returns a pair of narrowed values, both of type c, where c is supposed to be the least-upper bound in the sense of structural subtyping. The specification of lubNarrow once again illustrates the capability of OOHaskell to ‘program’ OO type-system aspects. We exploit the type-level reflection on HList records to define narrowing: instance ( HZip la va a , HZip lb vb b , HTIntersect la lb lc , H2ProjectByLabels lc a c aout , H2ProjectByLabels lc b c bout , HRLabelSet c ) => LubNarrow (Record a) (Record b) (Record c) where lubNarrow ra@(Record a) rb@(Record b) = ( hProjectByLabels lc ra , hProjectByLabels lc rb ) where lc = hTIntersect la lb (la,_) = hUnzip a (lb,_) = hUnzip b

That is, given two records ra and rb, we compute the intersection lc of their labels la and lb such that we can subsequently project both records to this shared label set. It is possible to improve consLub so that we can construct lists in linear time. We may also want to consider depth subtyping in addition to width subtyping, as we will discuss in Sec. 5.9.

5.2 Narrow to a fixed type The LUB narrowing is neither an explicit nor an implicit coercion. In the shapes example, we explicitly apply special list constructors, which we know perform coercion, but the target type is left implicit. Such semi-implicit narrowing is a feature of OOHaskell, not available in the otherwise similar systems OCaml and ML-ART. In OCaml, building the scribble list in the shapes example requires fully explicit narrowing (which OCaml calls upcast, “:>”):

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let (scribble: shape list) = [ (new rectangle 10 20 5 6 :> shape); (new circle 15 25 8 :> shape)] in ...

We can express such narrowing in OOHaskell as well: s1 <- mfix (rectangle (10::Int) (20::Int) 5 6) s2 <- mfix (circle (15::Int) 25 8) let scribble :: [Shape Int] scribble = [narrow s1, narrow s2]

The applications of narrow prepare the shape objects for insertion into the homogeneous Haskell list. We do not need to identify the target type per element: specifying the desired type for the result list is enough. The operation narrow is defined in a dedicated class: class Narrow a b where narrow :: Record a -> Record b

The operation narrow extracts those label-value pairs from a that are requested by b. Its implementation uses the same kind of projection on records that we saw in full in the previous section; cf. lubNarrow. (Fully) explicit narrowing implies that we must declare appropriate types — something that we managed to avoid so far. Here is the Shape type, which ‘drives’ narrowing in the example: -- The Shape interface type Shape a = Record ( :*: :*: :*: :*: :*: :*: :*:

GetX GetY SetX SetY MoveTo RMoveTo Draw HNil )

:=: :=: :=: :=: :=: :=: :=:

IO IO (a (a (a (a IO

a a -> -> -> -> ()

IO ()) IO ()) a -> IO ()) a -> IO ())

Two infix type synonyms add convenience to explicitly written types: infixr 2 :*: type e :*: l = HCons e l infixr 4 :=: type l :=: v = (l,v)

The Shape interface above explicitly includes the virtual operation draw because the loop over scribble needs this method. We will see more applications of narrow in subsequent sections. 5.3 Self-returning methods A self-returning method is a method whose result type is the type of self or is based on it. An example is a clone method. The typing of such methods (and of self) is known to be a difficult issue in typed object encodings; cf. (Cook et al., 1990; Abadi & Cardelli, 1996; Bruce et al., 2003) for some advanced treatments. In OOHaskell, we must not naively define a method me that returns self, as is:

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self_returning_point (x_init::a) self = do super <- printable_point x_init self returnIO $ me .=. self -- WRONG! .*. super

If we wrote such code, then we get a type error, when we attempt to instantiate the corresponding object (i.e., when we mfix the object generator). Haskell does not permit (equi-)recursive types, which are needed to type self in the example. The issue of recursive types and returning the full self is discussed in detail in Sec. 5.8. Here, we point out a simpler solution: disallowing returning self and requiring the programmer to narrow self to a specific desired interface. In the case of the clone method, mainstream programming languages typically define its return type to be the base class of all classes. The programmer is then supposed to use downcast to the intended subtype. We resolve the problem with the self-returning method as follows: self_returning_point (x_init::a) self = do super <- printable_point x_init self returnIO $ me .=. (narrow self :: PPInterface a) .*. super type PPInterface a = Record ( GetX :=: IO a :*: MoveX :=: (a -> IO ()) :*: Print :=: IO () :*: HNil )

That is, me narrows self explicitly to the interface for printable points. We should relate the explicit narrowing of the return type of me to the explicit declaration of the return type of all methods in C++ and Java. The presented narrowing approach does have a limitation however: all record components that do not occur in the target interface are irreversibly eliminated from the result record. We would prefer to make these components merely ‘private’ so they can be recovered through a safe downcast. We offer two options for such downcastable upcasts in the next two sections. 5.4 Casts based on dynamics Turning again to the shapes benchmark, let us modify the loop over scribble, a homogeneous list of shapes, so to single out circles for special treatment. This requires downcast: mapM_ (\shape -> maybe (putStrLn "Not a circle.") (\circ -> do circ # setRadius $ 10; circ # draw) ((downCast shape) ‘asTypeOf‘ (Just s2))) scribble

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In each iteration, we attempt to downcast the given shape object to the type of s2 (which we recall is a circle object). A downcast may fail or succeed, hence the Maybe type of the result. Neither OOHaskell’s narrow operation nor OCaml’s upcast support such scenarios. OOHaskell’s narrow irrevocably removes record components. We can define, however, other forms of upcast, which are reversible. We begin with a technique that exploits dynamic typing (Abadi et al., 1989; Abadi et al., 1991; L¨ammel & Peyton Jones, 2003). The new scribble list is built as follows: let scribble :: [UpCast (Shape Int)] scribble = [upCast s1, upCast s2]

where data UpCast x = UpCast x Dynamic

The data constructor UpCast is opaque for the library user, who can only upcast through a dedicated upCast operation. The latter saves the original object by embedding it into Dynamic. (We presume that record types readily instantiate the type class Typeable.) Dually, downcast is then a projection from Dynamic to the requested subtype: upCast :: (Typeable (Record a), Narrow a b) => Record a -> UpCast (Record b) upCast x = UpCast (narrow x) (toDyn x) downCast :: (Typeable b, Narrow b a) => UpCast (Record a) -> Maybe (Record b) downCast (UpCast _ d) = fromDynamic d

We want to treat ‘upcast objects’ as being objects too, and so we add a trivial HasField instance for looking up record components of upcast objects. This instance delegates the look-up to the narrowed part of the UpCast value: instance HasField l x v => HasField l (UpCast x) v where hLookupByLabel l (UpCast x _) = hLookupByLabel l x

This technique suffers from a few shortcomings. Although downcast is safe in a sense that no ‘bad things can happen’ (cf. unsafe casts in C), this downcast does not keep us from attempting so-called ‘stupid casts’, i.e., casts to types for which casting cannot possibly succeed. In the following section, we describe a more elaborate upcast/downcast pair that statically prevents stupid downcasts. The dynamics-based method also suffers from the full computational overhead of the narrow operation, a value-level coercion that iterates over all record components. 5.5 Casts based on unions We turn to the subtyping technique from Sec. 3.4 (further refined in Sec. 3.5), which used union types to represent the intersection of types. That techniques had several

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problems: it could not easily deal with the empty list, could not minimise the union type to the number of distinct element types, and could not downcast. We fully lift these restrictions here by putting type-level programming to work. We again make upcasts semi-implicit with dedicated list constructors: myOOP = do s1 <- mfix (rectangle (10::Int) (20::Int) 5 6) s2 <- mfix (circle (15::Int) 25 8) let scribble = consEither s1 (consEither s2 nilEither) ... and so on ...

The list constructors are almost identical to nilLub and consLub in Sec. 5.1. The difference comes when we cons to a non-empty list; see the last instance below: -- A type-level code for the empty list data NilEither -- The empty list constructor nilEither = ⊥ :: NilEither -- Cons as a trivial type-level function class ConsEither h t l | h t -> l where consEither :: h -> t -> l -- No coercion needed for a singleton list instance ConsEither e NilEither [e] where consEither h _ = [h] -- Construct union type for head and tail instance ConsEither e1 [e2] [Either e1 e2] where consEither h t = Left h : map Right t

We extend the union type for the ultimate element type with one branch for each new element, just as we did in the Haskell 98-based encoding of Sec. 3.5. However, with type-level programming, we can, in principle, minimise the union type so that each distinct element type occurs exactly once.14 This straightforward optimisation is omitted here for brevity.15 Method look-up is generic, treating the union type as the intersection of record fields of the union branches: instance (HasField => HasField where hLookupByLabel l hLookupByLabel l 14 15

l x v, HasField l y v) l (Either x y) v (Left x) = (Right y) =

hLookupByLabel l x hLookupByLabel l y

The same kind of constraints was covered in the HList paper (Kiselyov et al., 2004), cf. typeindexed heterogeneous collections. In essence, we need to iterate over the existing union type and use type-level type equality to detect if the type of the element to cons has already occurred in the union. If so, we also need to determine the corresponding sequence of Left and Right tags.

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Downcast is a type-driven search operation on the union type. We also want downcast to fail statically if the target types does not appear among the branches. Hence, we start downcast with a type-level Boolean, hFalse, to express that we have not yet seen the type in question: downCast = downCastSeen hFalse

Downcast returns Maybe because it can intrinsically fail at the value level: class DownCastSeen seen u s where downCastSeen :: seen -> u -> Maybe s

We process the union like a list. Hence, there are two cases: one for the nonsingleton union, and one for the final branch. Indeed, the details of the definition reveal that we assume right-associative unions. instance (DownCastEither seen b x y s, TypeEq x s b) => DownCastSeen seen (Either x y) s where downCastSeen seen = downCastEither seen (⊥::b) instance (TypeCastSeen seen b x s, TypeEq x s b) => DownCastSeen seen x s where downCastSeen seen = typeCastSeen seen (⊥::b)

In both cases we test for the type equality between the target type and the (left) branch type. We pass the computed type-level Boolean to type-level functions DownCastEither (‘non-singleton union’) and TypeCastSeen (‘final branch’, a singleton union): class TypeCastSeen seen b x y where typeCastSeen :: seen -> b -> x -> Maybe y instance TypeCast x y => TypeCastSeen seen HTrue x y where typeCastSeen _ _ = Just . typeCast instance TypeCastSeen HTrue HFalse x y where typeCastSeen _ _ = const Nothing

The first instance applies when we have encountered the requested type at last. In that case, we invoke normal, type-level type cast (cf. (Kiselyov et al., 2004)), knowing that it must succeed given the earlier check for type equality. The second instance applies when the final branch is not of the requested type. However, we must have seen the target type among the branches, cf. HTrue. Thereby, we rule out stupid casts. The following type-level function handles ‘non-trivial’ unions: class DownCastEither seen b x y s where downCastEither :: seen -> b -> Either x y -> Maybe s instance (DownCastSeen HTrue y s, TypeCast x s) => DownCastEither seen HTrue x y s where downCastEither _ _ (Left x) = Just (typeCast x) downCastEither _ _ (Right y) = downCastSeen hTrue y

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instance DownCastSeen seen y s => DownCastEither seen HFalse x y s where downCastEither _ _ (Left x) = Nothing downCastEither seen _ (Right y) = downCastSeen seen y

The first instances applies in case the left branch of the union type is of the target type; cf. HTrue. It remains to check the value-level tag. If it is Left, we are done after the type-level type cast. We continue the search otherwise, with seen set to HTrue to record that the union type does indeed contain the target type. The second instance applies in case the left branch of the union type is not of the target type; cf. HFalse. In that case, downcast continues with the tail of union type, while propagating the seen flag. Thereby, we rule out stupid casts. 5.6 Explicit type constraints In some cases, it is useful to impose structural record type constraints on arguments of an object generator, on arguments or the result type of a method. These constraints are akin to C++ concepts (Siek et al., 2005). The familiar narrow turns out to be a convenient tool for imposition of such type constraints. This use of narrow does no operations at run-time. A good example example of OO type constraints is the treatment of virtual methods in OOHaskell. Quoting from (Leroy et al. , 2004)[§ 3.4]: “It is possible to declare a method without actually defining it, using the keyword virtual. This method will be provided later in subclasses. A class containing virtual methods must be flagged virtual, and cannot be instantiated (that is, no object of this class can be created). It still defines type abbreviations (treating virtual methods as other methods.) class virtual abstract_point x_init = object (self) val mutable varX = x_init method print = print_int self#getX method virtual getX : int method virtual moveX : int -> unit end;;

In C++, such methods are called pure virtual and the corresponding classes are called abstract. In Java and C#, we can flag both methods and classes as being abstract. In OOHaskell, it is enough to leave the method undefined. Indeed, in the shapes example, we omitted any mentioning of the draw method when we defined the object generator for shapes. OCaml’s abstract point class may be transcribed to OOHaskell as follows: abstract_point x_init self = do xRef <- newIORef x_init returnIO $ varX .=. xRef .*. print .=. ( self # getX >>= Prelude.print ) .*. emptyRecord

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This object generator cannot be instantiated with mfix because getX is used but not defined. The Haskell type system effectively prevents us from instantiating classes which use the methods neither they nor their parents have defined. There arises the question of the explicit designation of a method as pure virtual, which would be of particular value in case the pure virtual does not happen to be used in the object generator itself. OOHaskell allows for such explicit designation by means of adding type constraints to self. To designate getX and moveX as pure virtuals of abstract_point we change the object generator as follows: abstract_point (x_init::a) self = do ... as before ... where _ = narrow self :: Record ( GetX :=: IO a :*: MoveX :=: (a -> IO ()) :*: HNil )

We use the familiar narrow operation, this time to express a type constraint. We must stress that we narrow here at the type level only. The result of narrowing is not used (cf. “ ”), so operationally it is a no-op. It does however affect the typechecking of the program: every instantiatable extension of abstract_point must define getX and moveX. One may think that the same effect can be achieved by adding regular type annotations (e.g., on self). These annotations however must spell out the desired object type entirely. Furthermore, a regular record type annotation rigidly and unnecessarily restrains the order of the methods in the record as well as their types (preventing deep subtyping, Sec. 5.9). One may also think object types can be simply constrained by specifying HasField constraints. This is impractical in so far that full object types would need to be specified then by the programmer; Haskell does not directly support partial signatures. Our narrow-based approach solves these problems.

5.7 Nominal subtyping In OCaml and, by default, in OOHaskell, object types engage into structural subtype polymorphism. Many other OO languages prefer nominal object types with explicitly declared subtyping (inheritance) relationships. There is an enduring debate about the superiority of either form of subtyping. The definite strength of structural subtype polymorphism is that it naturally enables inference of object types. The downside is potentially accidental subtyping (Cardelli & Wegner, 1985): a given object may be admitted as an actual argument of some function just because its structural type fits. Nominal types allow us to restrict subtyping polymorphism on the basis of explicitly declared subclass or inheritance relationships between nominal (i.e., named) types. Although OOHaskell is biased towards structural subtyping polymorphism,

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OOHaskell, as a general sandbox for typed OO language design, does admit nominal object types and nominal subtyping including multiple inheritance. We revisit our familiar printable points and colored points, switching to nominal types. First, we need to invent class names, or nominations: data PP data CP

= PP = CP

-- Printable points -- Colored points

As an act of discipline, we also register these types as nominations: class Nomination f instance Nomination PP instance Nomination CP

We attach nomination to a regular, record-based OOHaskell object as a phantom type. To this end, we using the following newtype wrapper: newtype N nom rec = N rec

The following two functions add and remove the nominations: -- An operation to ‘nominate’ a record as nominal object nominate :: Nomination nt => nt -> x -> N nt x nominate nt x = N x -- An operation to take away the type distinction anonymize :: Nomination nt => N nt x -> x anonymize (N x) = x

To be able to invoke methods on nominal objects, we need a HasField instance for N, with the often seen delegation to the wrapped record: instance (HasField l x v, Nomination f) => HasField l (N f x) v where hLookupByLabel l o = hLookupByLabel l (anonymize o)

OO programming with nominal subtyping on PP and CP can now commence. The object generator for printable points remains exactly the same as before except that we nominate the returned object as an PP: printable_point x_init s = do x <- newIORef x_init returnIO $ nominate PP -- Nominal! $ mutableX .=. x .*. getX .=. readIORef x .*. moveX .=. (\d -> modifyIORef x (+d)) .*. print .=. ((s # getX ) >>= Prelude.print) .*. emptyRecord

The nominal vs. structural distinction only becomes meaningful once we start to annotate functions explicitly with the requested nominal argument type. We will first consider request that insist on a specific nominal type, with no subtyping involved. Here is a print function that only accepts nominal printable points. printPP (aPP::N PP x) = aPP # print

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To demonstrate nominal subtyping, we define colored points (‘CP’): colored_point x_init (color::String) self = do super <- printable_point x_init self returnIO $ nominate CP -- Nominal! $ print .=. ( do putStr "so far - "; super # print putStr "color - "; Prelude.print color ) .<. getColor .=. (returnIO color) .*. anonymize super -- Access record!

We need to make CP a nominal subtype of PP. That designation is going to be explicit. We introduce a type class Parents, which is an extensible type-level function from nominal types to the list of their immediate supertypes. A type may have more than one parent: multiple inheritance. The following two instances designate PP as the root of the hierarchy and CP as its immediate subtype: class ( Nomination child, Nominations parents ) => Parents child parents | child -> parents instance Parents PP HNil instance Parents CP (HCons PP HNil)

-- PP has no parents -- Colored points are printable points

The OOHaskell library also defines a general relation Ancestor, which is the reflexive, transitive closure of Parents: class ( Nomination f, Nomination anc ) => Ancestor f anc

We are now in the position to define an upcast operation, which is the basis for nominal subtyping: -- An up-cast operation nUpCast :: Ancestor f g => N f x -> N g x nUpCast = N . anonymize

We could also define some forms of downcast. Our nUpCast does no narrowing, so operationally it is the identity function. This is consistent with the implementation of the nominal upcast in mainstream OO languages. The record type of an OOHaskell object is still visible in its nominal type. Our nominal objects are fully OOHaskell objects except that their subtyping is deliberately restricted. We can define a subtype-polymorphic print function for printable points by ‘relaxing’ the non-polymorphic printPP function through upcast.16 printPP (aPP::N PP x) = aPP # print -- accept PP only printPP’ o = printPP (nUpCast o) -- accept PP and nominal subtypes

The couple printPP and printPP’ clarifies that we can readily restrict argument types of functions to either precise types or to all subtypes of a given base. This granularity of type constraints it not provided by mainstream OO languages. Also, the use of structural subtyping in the body of printPP hints at the fact that we can blend nominal and structural subtyping with ease in OOHaskell. Again, this is beyond state-of-the-art in mainstream OO programming. 16

We cannot define printPP’ in a point-free style because of Haskell’s monomorphism restriction.

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In the previous section, we have studied nominal types for the sake of nominal subtyping. Nominal types are intrinsically necessary, when we need to model recursive object types in OOHaskell. In principle, a type system with equi-recursive types would be convenient in this respect. However, adding such types to Haskell was debated and then rejected because it will make type-error messages nearly useless (Hughes, 2002). Consequently, we encode recursive object types as iso-recursive types; in fact, we use newtypes. (An alternative technique of existential quantification (Pierce & Turner, 1994) is discussed in Sec. 5.11.) We illustrate iso-recursive types on uni-directionally linked dynamic lists. The interface of such list objects has methods that also return list objects: a getter for the tail and an insertion method. -- The nominal object type newtype ListObj a = ListObj (ListInterface a) -- The structural interface type type ListInterface a = Record ( IsEmpty :=: IO :*: GetHead :=: IO :*: GetTail :=: IO :*: SetHead :=: (a :*: InsHead :=: (a :*: HNil )

Bool a (ListObj a) -> IO ()) -> IO (ListObj a))

Recall that we had to define a HasField instance whenever we went beyond the normal ‘objects as records’ approach. This is the case here, too. Each newtype for iso-recursion has to be complemented by a trivial HasField instance: instance HasField l (ListInterface a) v => HasField l (ListObj a) v where hLookupByLabel l (ListObj x) = hLookupByLabel l x

For clarity, we chose the implementation of ListInterface a with two OO classes: for the empty and non-empty lists. A single OO list class would have sufficed too. Empty-list objects fail for all getters. Here is the straightforward generator for empty lists: nilOO self :: IO (ListInterface a) = returnIO $ isEmpty .=. returnIO True .*. getHead .=. failIO "No head!" .*. getTail .=. failIO "No tail!" .*. setHead .=. const (failIO "No head!") .*. insHead .=. reusableInsHead self .*. emptyRecord

The reusable insert operation constructs a new object of the consOO:

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reusableInsHead list head = do newCons <- mfix (consOO head list) returnIO (ListObj newCons)

Non-empty list objects hold a reference for the head, which is accessed by getHead and setHead. Here is the object generator for non-empty lists: consOO head tail self = do hRef <- newIORef returnIO $ isEmpty .=. .*. getHead .=. .*. getTail .=. .*. setHead .=. .*. insHead .=. .*. emptyRecord

head returnIO False readIORef hRef returnIO (ListObj tail) writeIORef hRef reusableInsHead self

OO programming on nominal objects commences without ado. They can be used just like record-based OOHaskell objects before. As an example, the following recursive function prints a given list. One can check that the various method invocations involve nominally typed objects. printList aList = do empty <- aList # isEmpty if empty then putStrLn "" else do head <- aList # getHead putStr $ show head tail <- aList # getTail putStr " " printList tail

5.9 Width and depth subtyping We have used the term subtyping in the informal sense of type-safe type substitutability. That is, we call the object type S to be a subtype of the object type T if in any well-typed program P the typeability of method invocations is preserved upon replacing objects of type T with objects of type S. This notion of subtyping is to be distinguished from behavioural subtyping, also known as Liskov Substitution Principle (Liskov & Wing, 1994). In OOHaskell, subtyping is enabled by the type of the method invocation operator #. For instance, the function \o -> o # getX has the following inferred type: HasField (Proxy GetX) o v => o -> v This type is polymorphic. The function will accept any object (i.e., record) o provided that it has the method labelled GetX whose type matches the function’s desired return type v.

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A basic form of subtyping or subsumption is width subtyping, whereupon an object of type S is a subtype of T if the record type S has (at least) all the fields of T with the exact same type. The HList library readily provides this subtyping relation, Record.SubType. Corresponding constraints can be added to type signatures (although we recall that Sec. 5.6 devised a constraint technique that is more convenient for OOHaskell). It is easy to see that if SubType S T holds for some record types S and T , then substituting an object of type S for an object of type T preserves the typing of every occurrence of # in a program. No method will be missing and no method will be of a wrong type. Width subtyping is only one form of subtyping. There are other subtyping relations, which too preserve the typing of each occurrence of # in a program — in particular, depth subtyping. While width subtyping allows the subtype to have more fields than the supertype, depth subtyping allows the fields of the subtype to relate to the fields of the supertype by subtyping. Typed mainstream OO languages like Java and C# do not support full depth subtyping. We will now explore depth subtyping in OOHaskell. We define some new object types and functions on the one-dimensional printable_point class from Sec. 4.2.5 and its extension colored_point from Sec. 4.3.3. We define a simple-minded onedimensional vector class, specified by two points for the beginning and the end, which can be accessed by the methods getP1 and getP2: vector (p1::p) (p2::p) self = do p1r <- newIORef p1 p2r <- newIORef p2 returnIO $ getP1 .=. readIORef .*. getP2 .=. readIORef .*. print .=. do self # self # .*. emptyRecord

p1r p2r getP1 >>= ( # print ) getP2 >>= ( # print )

The local type annotations p1::p and p2::p enforce our intent that the two points of the vector have the same type. It is clear that objects of type p must be able to respond to the message print. Otherwise, the type of the points is not constrained. Our object generator vector is parameterised over the class of points. In C++, the close analogue is a class template. This example shows that Haskell’s normal forms of polymorphism, combined with type inference, allow us to define parameterised classes without ado. We construct two vector objects, v and cv: testVector = do p1 <- mfix (printable_point 0) p2 <- mfix (printable_point 5) cp1 <- mfix (colored_point 10 "red") cp2 <- mfix (colored_point 25 "red") v <- mfix (vector p1 p2) cv <- mfix (vector cp1 cp2) -- ... to be continued ...

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The former is the vector of two printable points; the latter is the vector of two colored points. The types of v and cv are obviously different: the type checker will remind us of this fact if we tried to put both vectors into the same homogeneous list. The vectors v and cv are not related by width subtyping: indeed, both vectors agree on method names, but the types of the methods getP1 and getP2 differ. In v, the method getP1 has the type IO PrintablePoint whereas in cv the same method has the type IO ColoredPoint. These different result types, PrintablePoint and ColoredPoint, are related by width subtyping. The type of cv is a deep subtype of v. In OOHaskell, we may readily use functions (or methods) that exploit depth subtyping. For instance, we can define the following function for computing the norm of a vector, and we can pass either vector v or cv to the function. norm v = do p1 <- v # getP1; p2 <- v # getP2 x1 <- p1 # getX; x2 <- p2 # getX return (abs (x1 - x2))

The above test code continues thus: -- ... continued ... putStrLn "Length of v" norm v >>= Prelude.print putStrLn "Length of colored cv" norm cv >>= Prelude.print

The method invocation operations within norm remain well-typed no matter which vector, v or cv, we pass to that function. The typing of # is indeed compatible with both width and depth subtyping, and, in fact, their combination. Thus, the object type S is a subtype of T if the record type S has all the fields of T whose types are not necessarily the same but related by subtyping in turn. Here we assume, for now, that subtyping on method types is defined in accordance to conservative rules (Cardelli & Wegner, 1985; Abadi & Cardelli, 1996). (In the following formulation, without loss of generality, we assume that OOHaskell method types are monadic function types.) If A1 → · · · → An → IO R is a method type from T , then there must be a method type in S, with the same method name, and with a type A01 → · · · → A0n → IO R0 such that the following relationships hold: • A1 , . . . , An must be subtypes of A01 , . . . , A0n . • R0 must be a subtype of R.

(contra-variance) (co-variance)

The above vector example exercises the co-variance of the result type for the getters getP1 and getP2. We never had to specifically assert that the types of two objects are related by width or depth subtyping. This is because in each and every case, the compiler checks the well-typedness of all method invocations directly, so no separate subtyping rules are needed. We contrast this with type systems like System F≤ , where the subsumption rules are explicitly asserted. The only place where an OOHaskell

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programmer has to make the choice of subtyping relationship explicit is in explicit narrowing operations. The previously described operation narrow covers width subtyping; the OOHaskell library also includes an operation deep’narrow. For instance, we can place v and cv, into the same homogeneous list: let vectors = [v, deep’narrow cv]

The operation deep’narrow descends into records, prefixes method arguments by narrowing, and postfixes method results by narrowing. Deep narrowing is just another record operation driven by the structure of method types. (We refer to the source distribution for details.) Deep narrowing is not the only way of dealing explicitly with depth subtyping in OOHaskell. We may also adopt the union-type technique as of Sec. 5.5. 5.10 Co-variant method arguments The variance of argument types is the subject of a significant controversy (Castagna, 1995; Surazhsky & Gil, 2004; Howard et al., 2003). The contra-variant rule for method arguments entails type substitutability, i.e., it assures the type safety of method invocation for all programs. However, argument type contra-variance is known to be potentially too conservative. It is often argued that a co-variant argument type rule is more suitable for modelling real-world problems. If a method with co-variant argument types happens to receive objects of expected types, then co-variance is safe — for that particular program. The proponents of the co-variant argument type rule argue that because of the idiomatic advantages of the rule we should admit it for those programs where it is safe. It is the job of the compiler to warn the user when the co-variant rule is used unsafely. Alas, in the case of Eiffel — the most established language with co-variance — the situation is the following: “No compiler currently available fully implements these checks and behaviour in those cases ranges from run-time type errors to system crashes.” (comp.lang.eiffel, 2004). In this section we demonstrate the restrictiveness of contra-variance for methodargument types and show that OOHaskell’s subtyping naturally supports typesafe co-variance. The faithful implementation of the archetypal example from the Eiffel FAQ (comp.lang.eiffel, 2004) is contained in the accompanying source code. Continuing with the vector example from the previous section, we extend vector with a method, moveO for moving the origin of the vector. The method receives the new origin as a point object. vector1 (p1::p) (p2::p) self = do super <- vector p1 p2 self returnIO $ moveO .=. (\pa -> do p1 <- self # getP1 x <- pa # getX p1 # moveX $ x) .*. super

As in the previous section, we construct the vector1 of plain printable points

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v1 and the vector1 of colored points cv1. If we intend cv1 to be substitutable for v1 in all circumstances (by the virtue of depth subtyping), we must follow the contra-variance rule, which requires the argument pa of moveO be either a plain printable point (or an instance of its super-type). That requirement is responsible for the longer-than-expected implementation of moveO. Furthermore, the supertyping requirement on pa precludes moveO’s changing the color of the origin point, for the vector of colored points. That degrades the expressiveness. To illustrate the subtyping of the vectors, we define the function that moves the origin of its vector argument to 0: move_origin_to_0 varg = do zero <- mfix (printable_point 0) varg # moveO $ zero

We may indeed apply that function to either v1 or cv1. The function is polymorphic and can take any vector1 of plain points and and its subtypes. The type of cv1 is truly a deep subtype of the type of v1. (Again, OOHaskell does not require us to assert the relevant subtype relationship in any way.) We now turn to co-variant method argument types and so experiment with yet another class of vectors. We also construct two instances of vector2. vector2 (p1::p) (p2::p) self = do p1r <- newIORef p1 p2r <- newIORef p2 returnIO $ setO .=. writeIORef p1r -- ... other fields as in vector ... testVector = do -- ... test case as before ... v2 <- mfix (vectors p1 p2) -- vector of printable points cv2 <- mfix (vectors cp1 cp2) -- vector of colored points

Like vector1, vector2 provides for setting the origin point; cf. the method setO. However, vector2 does that in a direct and simple way; also, only vector2 permits changing the color of the origin point, in a vector of colored points. Although the method setO is more convenient and powerful than the method moveO, the method setO has co-variant argument types — across printable-point and colored-point vectors. For a vector of colored points, cv2, the argument type of setO must be a colored point too, i.e., the same type as p1r — otherwise, the mutation writeIORef cannot be typed. Hence, the type of cv2 cannot be a subtype of the type of v2 (because setO breaks the contra-variant argument type rule). An OO system that enforces the contra-variant rule will not allow us to write functions that can take both v2 and cv2. For example, we may want to devise the following function: align_origins va vb = do pa <- va # getP1 vb # setO $ pa

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It is always safe to apply align_origins to two vector2s of the same type. OOHaskell does let us pass either two vector2s of printable points (such as v2) to two vector2s of colored points (such as cv2), and so vector types can be substitutable — despite a co-variant argument type of set0. Substitutability is properly restricted for this function: set_origin_to_0 varg = do zero <- mfix (printable_point 0) varg # setO $ zero

We apply the function to v2, but if we try to apply it to cv2 we get the type error message about a missing method getColor (which distinguishes colored points from plain printable points). Likewise, we get an error if we attempt to place both v2 and cv2 in a homogeneous list like this: let vectors = [v2, deep’narrow cv2]

In this case, we can narrow both vectors to the type of vector though, so that the offending method setO will be projected out and becomes private. OOHaskell typechecks actual operations on objects; therefore, OOHaskell permits methods with co-variant argument types in situations where they are used safely. The type checker will flag any unsafe use and force the programmer to remove the offending method. Permitting safe uses of methods with co-variant argument types required no programming on our part. We get this behaviour for free.

5.11 Anti-patterns for subtyping We have seen several approaches to the construction of a subtype-polymorphic collection, as needed for the ‘scribble’ loop in the running shapes example. In the section on non-OOHaskell encodings, Sec. 3, we had discussed two additional options: • The use of HList’s heterogeneous lists. • The use of “∃” to make the list element type opaque. Albeit one might have expected these options to be of use, they turned out to be problematic for OO programming with non-extensible Haskell records. In the combination with OOHaskell (and its extensible records), these two options are even less attractive. In first approach, we construct the scribble list as: let scribble = s1 ‘HCons‘ (s2 ‘HCons‘ HNil)

and use hMap_, Sec. 3.7, to iterate over the list: hMapM_ (⊥::FunOnShape) scribble

where there must be an instance of type class Apply for FunOnShape, e.g.:

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instance ( HasField (Proxy Draw) r (IO ()) , HasField (Proxy RMoveTo) r (Int -> Int -> IO ()) ) => Apply FunOnShape r (IO ()) where apply _ x = do x # draw (x # rMoveTo) 100 100 x # draw

Haskell’s type class system requires us to provide proper bounds for the instance, hence the list of the method-access constraints (for “#”, i.e., HasField) above. The form of these constraints strongly resembles the method types listed in the shape interface type, Sec. 5.2. One may wonder whether we can somehow use the full type synonym Shape, in order to constrain the instance. This is not possible in Haskell because constraints are not first-class citizens in Haskell; we cannot compute them from types or type proxies — unless we were willing to rely on heavy encoding or advanced syntactic sugar. So we are doomed to manually infer and explicitly list such method-access constraints for each such piece of polymorphic code. The existential quantification approach falls short for essentially the same reason. Assuming a suitable existential envelope and following Sec. 3.6, we can build scribble as let scribble = [ HideShape s1, HideShape s2 ]

The declaration of the existential type depends on the function that we want to apply to the opaque data. When iterating over the list, via mapM_, we only need to unwrap the HideShape constructor prior to method invocations: mapM_ ( \(WrapShape shape) -> do shape # draw (shape # rMoveTo) 100 100 shape # draw ) scribble

These operations have to be anticipated in the type bound for the envelope: data OpaqueShape = forall x. ( HasField (Proxy Draw) x (IO ()) , HasField (Proxy RMoveTo) x (Int -> Int -> IO ()) ) => HideShape x

This approach evidently matches the HList-based technique in terms of encoding efforts. In both cases, we need to identify type class constraints that correspond to the (potentially) polymorphic method invocations. This is impractical. Not even mainstream OO languages with no advanced type inference, require this sort of type information from the programmer. Existential quantification can also be used for object encoding, e.g., for wrapping up self. That lets us, for example, easily implement self-returning methods without resorting to infinite types. Such use of existential quantification is not practical in OOHaskell for the same reason: it requires us to exhaustively enumerate all type classes an object and any of its types are or will be the instances of.

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We will first discuss usability issues of the current OOHaskell library, further constrained by current Haskell implementations. We will then summarise related work on functional object-oriented programming in Haskell and elsewhere. Finally, we will list topics for future work — other than just improving usability of OOHaskell. 6.1 Usability issues 6.1.1 Usability of inferred types So far, we have not shown any type inferred by Haskell for our objects. One may wonder how readable and comprehensible they are, if they can be used as means of program understanding, and if a Haskell language extension is needed to improve the presentation of the inferred types. In upshot, the inferred types are reasonable for simple OO programming examples, but there is a fuzzy borderline beyond which the volume and the idiosyncrasies of inferred types injure their usefulness. This concern suggests an important topic for future work. Let us see the inferred type of the colored point introduced in Sec. 4.3.1: ghci6.4> :t mfix $ colored_point (1::Int) "red" mfix $ colored_point (1::Int) "red" :: IO (Record (HCons (Proxy GetColor, IO String) (HCons (Proxy VarX, IORef Int) (HCons (Proxy GetX, IO Int) (HCons (Proxy MoveX, Int -> IO ()) (HCons (Proxy Print, IO ()) HNil))))))

We think that this type is quite readable, even though it reveals the underlying representation of records (as a heterogeneous list of label-value pairs), and gives away the proxy-based model for labels. We may hope for a future Haskell implementation whose customisable ‘pretty printer’ for types would present the result of type inference perhaps as follows: ghci> :t mfix $ colored_point mfix $ colored_point (1::Int) IO ( Record ( GetColor :=: :*: VarX :=: :*: GetX :=: :*: MoveX :=: :*: Print :=: :*: HNil ))

(1::Int) "red" "red" :: IO String IORef Int IO Int (Int -> IO ()) IO ()

The above example dealt with monomorphic objects. Let us also see the inferred type of a polymorphic object generator, with ‘open recursion left open’. Here is the (pretty-printed) type of the object generator for colored points: ghci> :t colored_point

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Num a HasField (Proxy GetX) r (IO a1) Show a1 => a -> String -> r -> IO ( Record ( GetColor :=: IO String :*: VarX :=: IORef a :*: GetX :=: IO a :*: MoveX :=: (a -> IO ()) :*: Print :=: IO () :*: HNil ))

The inferred type lists all the fields of an object, both new and inherited. Assumptions about self are expressed as constraints on the type variable r. The object generator refers to getX (through self), which entails a constraint of the form HasField (Proxy GetX) r (IO a1). The coordinate type for the point is polymorphic; cf. a for the initial value and a1 for the value retrieved by getX. Since arithmetics is performed on the coordinate value, this implies bounded polymorphism: only Num-ber types are permitted. We cannot yet infer that a and a1 must eventually be the same since ‘the open recursion is still open’. We must admit that we have assumed a relatively eager instance selection in the previous Haskell session. The Hugs implementation of Haskell is (more than) eager enough. The recent versions of GHC have become quite lazy. In a session with contemporary GHC (6.4), the inferred type would comprise the following additional constraints, which all deal with the uniqueness of label sets as they are encountered during record extension: HRLabelSet (HCons (HCons likewise for MoveX, likewise for MoveX, likewise for MoveX,

(Proxy (Proxy Print, Print, Print,

MoveX, a -> IO ()) Print, IO ()) HNil)), GetX GetX, VarX GetX, VarX, GetColor

Inspection of the HRLabelSet instances shows that these constraints are all satisfied, no matter how the type variable a is instantiated. No ingenuity is required. A simple form of strictness analysis were sufficient. Alas, GHC is consistently lazy in resolving even such constraints. Modulo HRLabelSet constraints, the inferred type seems quite reasonable, explicitly listing all relevant labels and types of the record components. 6.1.2 Usability of type errors Due to OOHaskell’s extensive use of type-class-based programming, there is a risk that type errors may become too complex. We will look at some examples. The results clearly provide incentive for future work on the subject of type errors. Let us first attempt to instantiate an abstract class, e.g., abstract_point from Sec. 5.6. That object generator defined the print method, which invoked getX on

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self. The latter is left to be defined in concrete subclasses. If we take the fixpoint of such an ‘incomplete’ object generator, Haskell’s type checker (here: GHC 6.4) gives the following error message: ghci> let x = mfix (abstract_point 7) No instance for (HasField (Proxy GetX) HNil (IO a)) arising from use of ‘abstract_point’ at :1:14-27 Probable fix: add an instance declaration for (HasField (Proxy GetX) HNil (IO a1)) In the first argument of ‘mfix’, namely ‘(abstract_point 7)’ In the definition of ‘x’: x = mfix (abstract_point 7)

We think that the error message is concise and to the point. The message succinctly lists just the missing field (The suggested ‘probable fix’ is not really helpful here). In our next scenario, we use a version of abstract_point that comprises an instantiation test by constraining self through narrow, as discussed in Sec. 5.6: abstract_point (x_init::a) self = do ... as before ... where _ = narrow self :: Record ( GetX :=: IO a :*: MoveX :=: (a -> IO ()) :*: HNil )

When we now take the fixpoint again, we get a more complex error message: ghci> let x = mfix (abstract_point 7) No instance for (HExtract HNil (Proxy GetX) (IO a), HExtract HNil (Proxy MoveX) (a -> IO ()), HasField (Proxy GetX) HNil (IO a1)) arising from use of ‘abstract_point’ at :1:14-27 Probable fix: ... In the first argument of ‘mfix’, namely ‘(abstract_point 7)’ In the definition of ‘x’: x = mfix (abstract_point 7)

Compared to the earlier error message, there are two additional unsatisfied HExtract constraints. Two out of the three constraints refer to GetX, and they complain about the same problem: a missing method implementation for getX. The constraint regarding MoveX deals with a pure virtual method that is not used in the object generator. The kinds and numbers of error messages for GetX and MoveX may lead to confusion; internals of OOHaskell end up at the surface. In order to improve on such problems, the Haskell type system and its errorhandling part would need to be opened up to allow for problem-specific error messages. We would like to refine Haskell’s type checker so that type error messages directly refer to the involved OO concepts. Let us consider yet another scenario. We turn to self-returning methods, as we discussed them in Sec. 5.3. In the following flawed OOHaskell program, we attempt to return self right away: self_returning_point (x_init::a) self = do

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super <- printable_point x_init self returnIO $ me .=. self -- assumes iso-recursive types .*. super

The problem will go unnoticed until we try to mfix the generator, at which point we get a type error: Occurs check: cannot construct the infinite type: a = Record (HCons (Proxy Me, a) (HCons (Proxy MutableX, IORef a1) (HCons (Proxy GetX, IO a1) (HCons (Proxy MoveX, a1 -> IO ()) (HCons (Proxy Print, IO ()) HNil))))) Expected type: a -> IO a Inferred type: a -> IO (Record (HCons (Proxy Me, a) (HCons (Proxy MutableX, IORef a1) (HCons (Proxy GetX, IO a1) (HCons (Proxy MoveX, a1 -> IO ()) (HCons (Proxy Print, IO ()) HNil)))))) In the application ‘self_returning_point 7’ In the first argument of ‘mfix’, namely ‘(self_returning_point 7)’

This error message is rather complex compared to the simple object types that are involved. Although the actual problem is correctly described, the programmer receives no help in locating the offending code, me .=. self. The volume of the error message is the consequence of our use of structural types. One may think that adding some type synonyms and using them in type signatures should radically improve the situation. It is true that contemporary Haskell type checkers keep track of type synonyms. However, an erroneous subexpression may just not be sufficiently annotated or constrained by its context. Also, the mere coding of type synonyms is very inconvenient. This situation suggests that a future Haskell type checker could go two steps further. Our first proposal is to allow for the inference of type synonyms; think of: foo x y z = ... -- complex expression on structural object types type Foo = typeOf foo -- capture the type in an alias

(Here, typeOf is an envisaged extension.) Our second proposal is to use type synonyms aggressively for the simplification of inferred types or type portions of error messages. This is a challenging subject given Haskell’s forms of polymorphism. The verbosity of OOHaskell error messages may occasionally compare to error messages in C++ template instantiation, which can be immensely verbose, spanning several dozens of packed lines, and yet boost and similar C++ libraries, which extensively use templates, are gaining momentum. In general, the clarity of error messages is undoubtedly an area that needs more research, and such research is being carried out by Sulzmann and others (Stuckey et al., 2004), which OOHaskell programmers and Haskell compiler writers may take advantage of. The ultimate conclusion of our discussion of inferred types and type errors is

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that such type information needs to be presented to the programmer in an abbreviated and OO-aware fashion. This proposal is based on the observation of OCaml’s development. Although objects types shown by OCaml are quite concise, that has not always been the case. In the ML-ART system, the predecessor of OCaml with no syntactic sugar (R´emy, 1994), the printed inferred types were not unlike the OOHaskell types we have seen in this section. “Objects have anonymous, long, and often recursive types that describe all methods that the object can receive. Thus, we usually do not show the inferred types of programs in order to emphasise object and inheritance encoding rather than typechecking details. This is quite in a spirit of ML where type information is optional and is mainly used for documentation or in module interfaces. Except when trying top-level examples, or debugging, the user does not often wish to see the inferred types of his programs in a batch compiler.”

6.1.3 Efficiency of object encoding Our representation of objects and their types is deliberately straightforward: polymorphic, extensible records of closures. This approach has strong similarities with prototype-based systems (such as Self (Ungar & Smith, 1987)) in that mutable fields and method ‘pointers’ are contained in one record. A more efficient representation based on separate method and field tables (as in C++ and Java) is possible, in principle. Although our current encoding is certainly not optimal, it is conceptually clearer. This encoding is used in such languages as Perl, Python, Lua — and is often the first one chosen when adding OO to an existing language. The efficiency of the current OOHaskell encoding is also problematic for reasons other than separation of fields and methods. For example, although record extension is constant (run-)time, the field/method lookup is linear search. Clearly, a more efficient encoding is possible: one representation of the labels in the HList paper permits a total order among the labels types, which in turn, permits construction of efficient search trees. We may also impose an order on the components per record type, complete with subtype-polymorphic record extension only to the right, so that labels can be mapped to array indexes. In the present paper, we chose conceptual clarity over such optimisations. Furthermore, a non-trivial case study is needed to drive optimisations. Mere improvements in object encoding may be insufficient however. The compilation time of OOHaskell programs and their runtime efficiency is challenged by the number of heavily nested dictionaries that are implied by our systematic type-class-based approach. It is quite likely that a scalable HList/OOHaskell style of programming will require compiler optimisations that make type-class-based programming more efficient — in general.

6.2 Related work Throughout the paper we referenced related work whenever specific technical aspects suggested to do so. We will complete the picture by a broader discussion.

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There are three overall dimensions of related work: foundations of object encoding (cf. Sec. 6.2.1), Haskell extensions for OO (cf. Sec. 6.2.2), and OO encoding in Haskell (cf. Sec. 6.2.3). The literature on object encoding is quite extensive. OOHaskell takes advantage of seminal work such as (Cardelli & Wegner, 1985; Abadi & Cardelli, 1996; Ohori, 1995; Pierce & Turner, 1994; Bruce & Mitchell, 1992). Most often, typed object encodings are based on polymorphic lambda calculi with subtyping, while there are also object calculi that start, more directly, from objects or records. Due to this overwhelming variety, we narrow down the discussion. We identify MLART (R´emy, 1994) by R´emy et al. (see also (R´emy & Vouillon, 1997)) as the closest to OOHaskell — in motivation and spirit, but not in the technical approach. Hence, Sec. 6.2.1 is entirely focused on ML-ART, without further discussion of less similar object encodings. The distinguishing characteristic of OOHaskell is the use of type-class-bounded polymorphism. 6.2.1 The ML-ART object encoding Both ML-ART and OOHaskell identify a small set of language features that make functional object-oriented programming possible. In both projects, the aim was to be able to implement objects — as a library feature. Therefore, several OO styles can be implemented, for different classes of users and classes of problems. One does not need to learn any new language and can discover OO programming progressively. Both ML-ART and OOHaskell base their object systems on polymorphic extensible records. Both OOHaskell and ML-ART deal with mutable objects (OOHaskell currently neglects functional objects since they are much less commonly used in practise). Both OOHaskell and ML-ART aim at preserving type inference. ML-ART adds several extensions to ML to implement objects: records with polymorphic access and extension, projective records, recursive types, implicit existential and universal types. As the ML-ART paper (R´emy, 1994) reports, none of the extensions are new, but their combination is original and “provides just enough power to program objects in a flexible and elegant way”. We make the same claim for OOHaskell, but using a quite different set of features. What fundamentally sets us apart from ML-ART is the different source language: Haskell. In Haskell, we can implement polymorphic extensible records natively rather than via an extension. We use type-class-based programming to this end.17 We avoid row variables and their related complexities. Our records permit introspection and thus let us implement various type-safe cast operations appealing to different subtyping relationships. For instance, unlike ML-ART, OOHaskell can compute the most common type of two record types without requiring type annotations. Quoting from the ML-ART paper: 17

The fact that such records are realisable in Haskell at all has been unknown, until the HList paper, which we published in 2004. The assumed lack of extensible records in Haskell was selected as prime topic for discussion at the Haskell 2003 workshop (H. Nilsson, 2003).

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“The same message print can be sent to points and colored points. However, both of them have incompatible types and can never be stored in the same list. Some languages with subtyping allow this set-up. They would take the common interface of all objects that are mixed in the list as the interface of any single object of the list.”

Unlike ML-ART, we do not rely on existential or implicitly universal types, nor recursive types. We use value recursion instead. That representation, a record of recursive closures, abstracts the internal state of the object — its value as well its type. Haskell helps us overcome what ML-ART calls “severe difficulties” with value recursion. In ML, the difficulties are serious enough to abandon the value recursion, despite its attractive features in supporting implicit subtyping, in favour of more complex object encodings requiring extensions of the type system. The subtle problem of value recursion is responsible for complicated and elaborate rules of various mainstream OO languages that prescribe what an object constructor may or may not do. The ML-ART paper mentions an unpublished attempt (by Pierce) to take advantage of the facts that fixpoints in a call-by-name language are always safe and that call-by-name can be emulated in a call-by-value language with the help of extra abstraction (thunks). However, in that attempted implementation the whole message table had to be rebuilt every time an object sends a message to self and so that approach was not pursued further. Our simple scheme of Sec. 4.4 seems to answer the ML-ART challenge — “to provide a clean and efficient solution that permits restricted form of recursion on non-functional values.” ML-ART uses a separate method table, whereas OOHaskell uses a single record for both mutable fields and method ‘pointers’. The ML-ART encoding is more efficient than that of OOHaskell. All instances of an object (class) literally share the same method table. ML-ART (and OCaml) is also more efficient simply because more elements of the object encoding are natively implemented. By contrast, OOHaskell’s type system is programmed through type-class-based programming. As a result, OOHaskellis definitely less fit for practical OO software development than ML-ART (or rather OCaml). 6.2.2 Haskell language extensions There were attempts to bring OO to Haskell by a language extension. An early attempt is Haskell++ (Hughes & Sparud, 1995) by Hughes and Sparud. The authors motivated their extension by the perception that Haskell lacks the form of incremental reuse that is offered by inheritance in object-oriented languages. Our approach uses common extensions of the Hindley/Milner type system to provide the key OO notions. So in a way, Haskell’s fitness for OO programming just had to be discovered, which is the contribution of this paper. O‘Haskell (Nordlander, 1998; Nordlander, 2002) is a comprehensive OO variation on Haskell designed by Nordlander. O‘Haskell extends Haskell with reactive objects and subtyping. The subtyping part is a substantial extension. The reactive object part combines stateful objects and concurrent execution, again a major extension. Our development shows that no extension of Haskell is necessary for stateful objects, and the details of the object system can be programmed in Haskell.

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Another relevant Haskell variation is Mondrian. In the original paper on the design and implementation of Mondrian (Meijer & Claessen, 1997), Meijer and Claessen write: “The design of a type system that deals with subtyping, higherorder functions, and objects is a formidable challenge ...”. Rather than designing a very complicated language, the overall principle underlying Mondrian was to obtain a simple Haskell dialect with an object-oriented flavour. To this end, algebraic datatypes and type classes were combined into a simple object-oriented type system with no real subtyping, with completely co-variant type-checking. In Mondrian, runtime errors of the kind “message not understood” are considered a problem akin to partial functions with non-exhaustive case discriminations. OOHaskell raises the bar by providing proper subtyping (“all message will be understood”) and other OO concepts in Haskell without extending the Haskell type system.

6.2.3 Object encodings for Haskell This paper may claim to provide the most authoritative analysis of possible object encodings in Haskell; cf. Sec. 3. Previous published work on this subject has not addressed general (functional) object-oriented programming, but it has focused instead on the import of foreign libraries or components into Haskell (Finne et al., 1999; Shields & Peyton Jones, 2001; Pang & Chakravarty, 2004). The latter problem domain makes important simplifying assumptions: • • • • • •

Object state does not reside in Haskell data. There are only (opaque) object ids referring to the foreign site. State is solely accessed through methods (“properties”). Haskell methods are (often generated) stubs for foreign code. As a result, such OO styles just deal with interfaces. No actual (sub)classes are written by the programmer.

In this restricted context, one approach is to use phantom types for recording inheritance relationships (Finne et al., 1999). Each interface is represented by an (empty) datatype with a type parameter for extension. After due consideration, it turns out that this approach is a restricted version of what Burton called “type extension through polymorphism”: even records can be made extensible through the provision of a polymorphic dummy field (Burton, 1990). Once we do not maintain Haskell data for objects, there is no need to maintain a record type, but the extension point is a left over, and it becomes a phantom. We have “re-generalised” the phantom approach in Sec. 3.2. Another approach is to set up a Haskell type class to represent the subtyping relationship among interfaces (Shields & Peyton Jones, 2001; Pang & Chakravarty, 2004) where each interface is modelled as a dedicated (empty) Haskell type. We have enhanced this approach by state in Sec. 3.5. Based on our detailed analysis of both approaches, we submit that the second approach seems to be slightly superior to the first one, while both approaches are too cumbersome for actual functional OO programming.

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Not in peer-referred publications, but in Haskell coding practise, some sorts of OO-like encodings are occasionally found. For instance, it is relatively well understood that Haskell’s type classes allow for interface polymorphism or for abstract classes (type classes) vs. concrete classes (type class instances). As of writing, the published Haskell reference solution for the shapes example, http: //www.angelfire.com/tx4/cus/shapes/, is a simple-to-understand encoding that does not attempt to maximise reuse among data declarations and accessors. The encoding is specialised to the specific problem; the approach may fail to scale. The encoding also uses existentials for handling subtype-polymorphic collections, which is an inherently problematic choice, as we have shown in Sec. 5.11. 6.3 More future work We have focused on mutable objects so far; studying functional objects appears to be a natural continuation of this work, even though functional objects are of much less practical relevance. The notion of object construction as a multi-stage computation (cf. Sec. 4.4) merits further exploration (as well as the clarification of the relationship with environment classifiers (Taha & Nielsen, 2003)). OOHaskell should be elaborated to cover general forms of reflective programming and, on the top of that, general forms of aspect-oriented programming. A simple form of reflection is already provided in terms of the type-level encoding of records. We can iterate over records and their components in a generic fashion. Further effort is needed to cover more advanced forms of reflection such as the iteration over the object pool, or the modification of object generators. Another promising elaboration of OOHaskell would be its use for the reusable representation of design-pattern solutions. 7 Concluding remarks The present paper addresses the intellectual challenge of seeing if the conventional OO idioms can at all be implemented in Haskell (short of writing a compiler for an OO language in Haskell). Peyton Jones and Wadler’s paper on imperative programming in Haskell (Peyton Jones & Wadler, 1993) epitomises such an intellectual tradition for the imperative paradigm. The same kind of intellectual challenge, ‘paradigm assimilation’, is addressed by FC++ (McNamara & Smaragdakis, 2004), which implements in C++ the quintessential Haskell features: type inference, higher-order functions, non-strictness. The present paper, conversely, faithfully (i.e., in a similar syntax and without global program transformation) realises a principal C++ trait — OO programming. According to Peyton Jones, Haskell is “the world’s finest imperative programming language” (Peyton Jones, 2001). We submit that Haskell is also a bleeding-edge OO programming language, while we readily restrict this claim to mere OO language-design capability; much more work would be needed to enable scalable OO software development with Haskell. We have discovered an object system for Haskell that supports stateful objects,

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inheritance and subtype polymorphism. We have implemented OO as a Haskell library, OOHaskell, based on the polymorphic, extensible records with introspection and subtyping provided by the HList library (Kiselyov et al., 2004). Haskell programmers can use OO idioms if it suits the problem at hand. We have demonstrated that OOHaskell programs are very close to the textbook OO code, which is normally presented in mainstream OO languages. OOHaskell’s deviations are appreciated. The OOHaskell library offers a comparatively rich combination of OO idioms. Most notably, we have implemented parameterised classes, constructor methods, abstract classes, pure virtual methods, single inheritance, multiple inheritance, object composition, structural types, and nominal types. The choice of Haskell as a base language has allowed us to deliver extensive type inference, firstclass classes, implicit polymorphism of classes, and more generally: programmable OO type systems. Starting from the existing OOHaskell library and the corresponding sample suite, one can explore OO language design, without the need to write a compiler. The present paper settles the question that hitherto has been open. The conventional OO idioms in their full generality are expressible in current Haskell without any new extensions. It turns out, Haskell 98 plus multi-parameter type classes with functional dependencies are sufficient. This combination is well-formalised and reasonably understood (Stuckey & Sulzmann, 2005). Even overlapping instances are not essential (yet using them permits a more convenient representation of labels, and a more concise implementation of some type-level functionality). The fact that we found a quite unexpected (and unintended) use of the existing Haskell features is reminiscent of the accidental discovery of C++ template meta-programming. The latter is no longer considered an exotic accident or a type hack — rather, a real feature of the language (Czarnecki et al., 2003), used in the Standard Template Library and described in popular C++ books, e.g., (Alexandrescu, 2001). Haskell has let us move beyond the mere curiosity of implementing OO idioms to the point of making contributions to open and controversial OO problems. Haskell has let us concisely specify and enforce the restrictions on the behaviour of object constructors (preventing the constructor access not-yet-fully constructed objects). The object encoding with recursive records can be made safe. Also, we were able to effortlessly implement fine-grain notions of width and depth subtyping, with respect to particular object operations, and thus safely permit methods with co-variant argument subtyping. Not only OOHaskell is able to automatically compute the least general interface of a heterogeneous collection of objects (through semi-implicit upcasts) and make the collection homogeneous, but it provides the means for safe downcasts. Moreover, downcasts that cannot possibly succeed are flagged as type errors. These are capabilities that go beyond state-of-the-art functional objectoriented programming with OCaml. Just as C++ has become the laboratory for generative programming (Czarnecki et al., 2003) and lead to such applications as FC++ (McNamara & Smaragdakis, 2004) and Boost (http://www.boost.org/), we contend that (OO)Haskell would fit as the laboratory for advanced and typed OO language design. All our exper-

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iments have shown that (OO)Haskell indeed supports a good measure of experimentation — all without changing the type system and the compiler.

Acknowledgements We thank Keean Schupke for his major contributions to the HList and OOHaskell libraries. We thank Chung-chieh Shan for very helpful discussions. We also gratefully acknowledge feedback from Robin Green, Bryn Keller, Chris Rathman and several other participants in mailing list or email discussions. The second author presented this work at an earlier stage at the WG2.8 meeting (Functional Programming) in November 2004 at West Point. We are grateful for feedback received at this meeting.

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